According to a witness on the northern Gaza border, "around 25 Israeli tanks crossed the border in waves of eight tanks" at dawn today.
A military source told AFP "a limited number of troops entered the northern Gaza Strip to conduct searches for explosive devices and tunnels" dug by militants into Israel.
Israel struck at the heart of the Palestinian government on Sunday, hitting the office of prime minister Ismail Haniya, warning it would use all its power to free a soldier captured by Palestinian militants a week ago.
The armed wing of Hamas responded to the strike by threatening to resume attacks inside Israel, on power plants, institutions and schools.
“The Ezzedine al-Qassam Brigades warn the Zionist enemy: if its operations continue, we will hit the occupation targets we were previously reluctant to strike," said a statement received by news agency AFP in Gaza City.
The Ezzedine Brigades have carried out suicide bombings in Israel over the past decade although it has not carried out any such attacks for more than a year.
But Israeli Prime Minister Ehud Olmert vowed his government would not bow to "blackmail" as Palestinians cautioned that Egyptian-led mediation efforts under way to free the captured 19-year-old corporal, Gilad Shalit, were faltering.
"Efforts continue but so far in vain. We are near an impasse," Palestinian leader Mahmoud Abbas's spokesman Nabil Abu Rudeina told reporters.
Israel sent troops back into the Gaza Strip last week launching waves of night air raids in its biggest military operation in a year.
More air strikes
In the latest air strike carried out in the early hours of Monday morning a Palestinian was wounded in an air strike against Hamas offices in Gaza City while three others were wounded by shrapnel from Israeli artillery fire near the northern Gaza town of Beit Lahiya.
The Israeli military said that it had carried out two air raids targeting Gaza City buildings run by the Al-Aqsa Martyrs Brigades militant group, which is loosely affiliated to Palestinian president Mahmud Abbas's Fatah movement.
The third target in the nearby town of Beit Hanun whjich the Israeli army claimed was a weapons depot used for "producing and storing" makeshift rockets.
Haniya office shelled
Meanwhile witnesses said Israeli helicopter gunships fired on Mr Haniya’s office shortly before 2:00 am on Sunday (2300 GMT Saturday), setting the building ablaze.
"It's an attack against a Palestinian symbol," said Mr Haniya, who was not in the office at the time.
"We ask the international community and the Arab League to take its responsibilities towards our people and intervene" to end what he called Israel's "insane policy."
Mr Abbas, inspecting the damage done to the prime minister's office, lashed out at Israel.
"Destroying institutions for the Palestinian people, targeting a power plant or the office of prime minister Haniya are truly criminal operations," he said.
The pre-dawn strike on Haniya’s office followed warnings from Israel that no-one in the Hamas-led government is immune from attack.
Israeli Deputy Prime Minister Shimon Peres insisted the raid on Haniya's office was a shot across the government's bows rather than an assassination bid.
"It was not an attempt on his life, but it was a clear warning to stop this double behaviour," Mr Peres told CNN.
A second air strike in the northern Gaza Strip town of Jabaliya killed a member of the Hamas armed wing and wounded another.
An Israeli army spokesman said three more militants were shot dead in the southern Gaza Strip. It was not immediately clear to which faction they belonged.
Crisis escalating
Palestinian chief negotiator Saeb Erakat warned that the Israeli air strikes were only escalating the crisis.
"I'm really afraid that every hour that passes ... we're going to lose the ability to solve the crisis, diplomatically and politically," he told CNN.
And UN Secretary General Kofi Annan expressed concern that the Israeli onslaught could undermine the possibilities for a lasting settlement of the Middle East conflict.
"I remain very concerned about the need to preserve Palestinian institutions and infrastructure -- they will be the basis for an eventual two-state solution and that's in the interest of both Israel and the Palestinians," the UN chief said.
"It would therefore seem inadvisable to carry out action that will have the opposite effect."
Israel has rejected outright the demands of militant groups which seized Corporal Shalit in an attack on an army post on the Gaza border on June 25 and are now seeking the release of Palestinian prisoners.
A Palestinian official has quoted mediators as saying 19-year-old Shalit is alive after being treated for wounds.
Israel has also threatened to strike at Hamas leaders, including those based in Damascus, raising fears of a regional escalation of the worst crisis in the Middle East since Hamas came to power and Mr Olmert took the helm in Israel.
With the threat of a full scale Israeli ground offensive looming, already impoverished residents of Gaza are grappling with shortages of food, fuel and electricity.
Israel temporarily opened the Karni border crossing into Gaza to allow in humanitarian supplies including food, and it resumed pumping fuel.