Israeli aircraft have bombed a Hamas security target in Gaza City and tunnels used by the militant group along the border with Egypt.
There were no reports of casualties after the strikes late on Sunday.
Israeli PM Ehud Olmert earlier vowed a "disproportionate" response to rocket attacks on Israel from Gaza.
He was speaking soon after at least two rockets hit southern Israel. Later three Israelis were injured in a mortar barrage by Palestinian militants.
Late on Sunday, Israeli aircraft carried out a strike on an empty Hamas police station in Gaza City, Palestinian witnesses say.
They say the building had been vacated after an Israeli warning to residents to leave the area.
Palestinians also reported explosions in Rafah on the Gaza-Egypt border, where Hamas operates tunnels to smuggle in weapons and food.
The Israeli army has so far made no comments.
Abbas criticism
Two weeks ago, Israel and Hamas - which controls Gaza - declared separate truces ending a three-week conflict.
Palestinian Authority and Hamas officials are gathering in Cairo for talks aimed at boosting the ceasefires.
An adviser to Ismail Haniya, who heads the Hamas government in Gaza, told AFP news agency the militant group was waiting for Israel's response to a truce offer, transmitted by Egypt, adding that things were "moving in a positive direction".
The Egyptians have been leading efforts to broker a permanent ceasefire by holding separate talks with officials from Israel and Hamas.
But Palestinian leader Mahmoud Abbas told journalists in Cairo that talks were impossible with anyone who rejected the supremacy of the Palestine Liberation Organisation - in an apparent reference to Hamas's leadership.
He also accused Hamas of having "taken risks with the blood of Palestinians, with their fate, and dreams and aspirations for an independent Palestinian state".
Civilian casualty
Israel wants the rocket attacks to end and wants to prevent militants in Gaza from being able to rearm.
Hamas wants the border crossings into Gaza to be fully opened to end a 18-month blockade of Gaza.
Speaking at the weekly Israeli cabinet meeting, Mr Olmert warned Israel would respond forcefully to renewed rocket fire.
"We've said that if there is rocket fire against the south of the country, there will be a disproportionate Israeli responce to the fire on the citizens of Israel and its security forces," he said.
"The response will come at the time, the place and the manner that we choose."
His strong stance was echoed by Foreign Minister Tzipi Livni and Defence Minister Ehud Barak.
Soon afterwards, militants fired mortars at an Israeli village near the Gaza border, injuring two soldiers and a civilian.
The civilian casualty is the first since the 18 January ceasefires which ended Israel's three-week assault on Gaza, which it said was aimed at stopping rocket attacks.
Earlier, a rocket landed between two nursery schools in Eshkol region of southern Israel, media reports said.
On Saturday, a rocket fired from Gaza exploded near the Israeli city of Ashkelon, with no casualties reported, and at least two were fired in the days before.
The ceasefires, independently declared by each side, have been violated several times.
An Israeli soldier was killed in a bomb attack on the Gaza border last Tuesday. Israel responded with air raids and a brief ground incursion by soldiers and tanks.
About 1,300 Palestinians and 10 Israeli soldiers were killed in Israel's devastating three-week assault on Gaza. Three Israeli citizens died in rocket attacks.
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Turkey's prime minister has stormed off the stage at the World Economic Forum in Davos after a heated debate on Gaza with Israel's president.
Recep Tayyip Erdogan clashed with Shimon Peres, whose voice had risen as he made an impassioned defence of Israel's actions, jabbing his finger.
Mr Erdogan said Mr Peres had spoken so loudly to conceal his "guilt".
He accused the moderator of not allowing him to speak and said he did not think he would return to Davos.
The Turkish PM stressed later that he had left the debate not because of his disagreements with Mr Peres but because he had been given much less time to speak than the Israeli leader.
Turkey is one of the few Muslim countries to have dealings with Israel, but relations have been under strain since the Islamist-rooted AK Party was elected to power in 2002.
Late on Thursday, a WEF official said that Mr Peres and Mr Erdogan had spoken by mobile telephone, and both men now considered the matter closed.
Dinner time
In the debate, Mr Erdogan was cut off as he attempted to reply to Mr Peres.
Earlier the Turkish Prime Minister had made an address himself, describing Gaza as an "open-air prison".
When the audience applauded Mr Peres, he said: "I find it very sad that people applaud what you said. You killed people. And I think that it is very wrong."
The moderator, Washington Post columnist David Ignatius, had given him a minute to reply, then asked him to finish, saying that people needed to go to dinner.
"I do not think I will be coming back to Davos after this because you do not let me speak," Mr Erdogan shouted before marching off the stage in front of Mr Peres, UN Secretary General Ban Ki-moon and an elite audience of ministers and international officials.
Mr Peres had told the audience Israel was forced on to the offensive against Hamas by thousands of rockets and mortars fired into Israel.
"The tragedy of Gaza is not Israel, it is Hamas," the Israeli leader said.
"Why did they fire rockets? There was no siege against Gaza. Why did they fight us, what did they want? There was never a day of starvation in Gaza."
He argued that Mr Erdogan would have reacted in the same way if rockets had hit Istanbul.
More than 1,300 Palestinians and 14 Israelis were killed during the three-week conflict which began on 27 December.
At news conference later on Thursday, Mr Erdogan complained that he had been allowed to speak just 12 minutes to 25 for Mr Peres.
"I did not target at all in any way the Israeli people, President Peres or the Jewish people," he said.
"I am a prime minister, a leader who has specifically expressly stated that anti-Semitism is a crime against humanity."
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The UN will launch an appeal for $613m to help people affected by Israel's military offensive in Gaza.
"These needs are massive and multi-faceted," the body's top official Ban Ki-Moon is quoted saying at the World Economic Forum in Davos.
He added that funds would be used to "help overcome at least some measures of this hardship".
The announcement came amid fears of a collapse of unilateral ceasefires by the Israel army and Hamas militants.
Two rockets have been fired at Israel, while air strikes hit southern Gaza.
UN Secretary-General Mr Ban visited Gaza after the ceasefires; he said help was urgently needed and he had been deeply moved by what he saw.
The appeal is reported to cover requirements of the UN and other aid organisations for the next six to nine months and will provide everything from medical care to clean water.
He told reporters that an appeal for longer-term needs will be launched later.
The Israeli offensive in Gaza killed about 1,300 Palestinians, of whom 412 were children; 21,000 homes were destroyed or badly damaged. Thirteen Israelis were killed.
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Israel mounts fresh attacks as international ceasefire efforts gather pace
George Bush today blamed the continuing violence in Gaza on Hamas terrorism and offered no criticism of Israel in his first comments since Israeli air strikes began a week ago.
The US president condemned Hamas's campaign of rocket attacks on Israel as an "act of terror" and said no peace deal would be acceptable unless the flow of smuggled weapons to terrorist groups was monitored and stopped.
"This recent outburst of violence was instigated by Hamas, a Palestinian terrorist group supported by Iran and Syria that calls for Israel's destruction," he said.
Bush said Hamas ended the latest ceasefire on 19 December and "soon unleashed a barrage of rockets and mortars that deliberately targeted innocent Israelis, an act of terror that is opposed by the legitimate leader of the Palestinian people, President [Mahmoud] Abbas".
Israeli aircraft and gunboats continued to target Hamas in Gaza today. In the latest attacks, the Israelis struck the homes of two Hamas operatives, saying the buildings were used to store weapons and plan attacks. Hamas said one of its military leaders, Abu Zakari al-Jamal, was killed in an air raid last night.
More than 400 Palestinians and at least four Israelis have been killed since Israel launched its offensive last Saturday. The UN estimated yesterday that a quarter of the Palestinians killed were civilians.
The White House will not comment on whether it views the Israeli response as proportionate.
In his weekly radio address, Bush said Washington was "leading diplomatic efforts to achieve a meaningful ceasefire that is fully respected".
He said he would keep the president-elect, Barack Obama, updated on the negotiations. Obama has refused to comment on the crisis while Bush remains in office.
The UN secretary general Ban Ki-moon last night urged world leaders to intensify efforts to achieve an immediate ceasefire.
Israel has said it will accept a ceasefire only if it is enforced by international monitors, and Bush backed that stance by saying a truce that involved continued rocket attacks would not be acceptable.
"Promises from Hamas will not suffice. There must be monitoring mechanisms in place to help ensure that smuggling of weapons to terrorist groups in Gaza comes to an end," he said.
He expressed concern about the humanitarian suffering of the Palestinian people in Gaza, but again blamed this on Hamas.
"By spending its resources on rocket launchers instead of roads and schools, Hamas has demonstrated that it has no intention of serving the Palestinian people," Bush said.
He also condemned Hamas for hiding within the civilian population. "Regrettably, Palestinian civilians have been killed in recent days," he said.
Abbas, the moderate leader of the West Bank, and several Arab foreign ministers are flying to New York this weekend to urge the UN security council to adopt an Arab draft resolution that would condemn Israel and demand a halt to its bombing campaign in Gaza.
Meanwhile, Israeli troops remain massed on the border, waiting for the signal to invade Gaza. Speaking from Syria, the exiled Hamas leader, Khaled Mashaal, warned that any ground assault would lead Israel to "a black destiny of dead and wounded".
Palestinian militants continued to fire rockets into southern Israel today. There were no injuries, though one rocket hit a house in the southern city of Ashkelon, police said.
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