A 19-year-old soldier, who also holds French nationality, went missing when the Popular Resistance Committees attacked an army post on the Gaza Strip border at dawn on Sunday.
Hamas has come under increased pressure over the incident as Israel masses tanks on the border of the Gaza Strip.
The inner cabinet of Ehud Olmert, the Israeli prime minister, gave the Palestinians 48 hours to return Corporal Gilad Shalit.
Olmert said on Monday that he held Hamas and Mahmoud Abbas, the Palestinian president, responsible for the raid.
One source, speaking on condition of anonymity, told AFP that Israel would work to ensure the fall of the Hamas-led government if Shalit were not released.
"We will make sure that the Hamas government ceases to operate if the kidnapped soldier is not returned to us alive," the source said.
Yuval Diskin, the head of Israel's Shin Beth homeland security agency, made the threat in talks with Abbas late on Sunday, the source said.
Haim Ramon, the Israeli justice minister, told Israel Radio: "We hold Abu Mazen [Abbas] responsible. He holds ultimate responsibility for what goes on."
Two other Israeli servicemen and two Palestinian fighters also died in the attack.
Later on Monday, Abbas ordered his security services to search for the soldier, an official said.
An Israeli border alert preceded
the bloody clash and kidnapping
"President Mahmud Abbas ordered prime minister Ismail Haniya, interior minister Said Siam and security service commanders to immediately launch a serious search today to release the soldier," the source told AFP.
Egyptian emissaries were helping to defuse the crisis, which has called into question Olmert's plan to withdraw unilaterally from parts of the West Bank, the other territory the Palestinians want as part of a state.
The source said the manhunt would take place "throughout the Gaza Strip", and that the security services were about to deploy.
"We are continuing our efforts to release the kidnapped soldier. As of now, we have been told that the soldier is fine. He is in good condition and he is being treated well"
Ghazi Hamad, spokesman for the government, confirmed to Israeli army radio that a number of parties, including the Egyptians, were working to resolve the stand-off.
France also said it was in contact "with all the concerned parties to find a solution to this situation".
There were conflicting reports from the Popular Resistance Committees (PRC) group, which carried out the mission, as to whether it was holding the soldier.
At one stage, the group denied knowing his whereabouts, but later said he was alive, in good health and not seriously injured.
A Palestinian security official, speaking on condition of anonymity, said: "We are continuing our efforts to release the kidnapped soldier. As of now, we have been told that the soldier is fine. He is in good condition and he is being treated well."
The PRC, the armed wing of Hamas and the previously unheard of Army of Islam, have called for the release of all Palestinian women and children from Israeli jails in exchange for information on Shalit.
Olmert ruled out a prisoner exchange for the Israeli army conscript. "There will be no negotiations, no bargaining, no agreements," he said.
Ramon said Israel believed that the soldier was alive, though he might have been wounded.
Under pressure from Israel's opposition parties, which do not want to cede occupied land to the Palestinians, Olmert has pursued action against Gaza fighters despite an outcry over the killing of 14 Palestinian civilians, including children, in recent Israeli air strikes.
Israeli officials have hinted that, should the deadline for Shalit's release go unmet, there could be an aerial attack on Gaza against the civilian infrastructure and Hamas leaders including Ismail Haniya, the Palestinian prime minister.
Meanwhile, an armed group loosely affiliated to Abbas's Fatah party has urged the kidnappers not to release the soldier.
"We ask our brother fighters not to release the Zionist soldier regardless of all the pressures and threats," Al-Aqsa Martyrs Brigades said in a statement on Monday.
Ready to confront
The group said its forces were ready to confront any Israeli offensive in the Gaza Strip.Hamas's armed wing was among three resistance groups that said Sunday's assault was to avenge the death of 22 Palestinian civilians killed in one alleged Israeli shelling and a series of air strikes.
Nasseredine al-Shaer, deputy prime minister, has demanded the immediate release of the soldier and Abbas also condemned the attack.
Outside the Kerem Shalom gate that separates Israel from Gaza, the army was busy positioning tanks, ground troops and artillery, waiting for the green light from political leaders to proceed.