Israeli Prime Minister Ehud Olmert rejected any negotiations with the militants, and the army pressed ahead with its Gaza offensive. Privately, though, some Israeli officials said the government had not ruled out any options to win Cpl. Gilad Shalit's freedom.
Israel has pounded Gaza with air strikes and artillery shells for nearly a week in an unsuccessful effort to force the Hamas-linked militants to release Shalit. It has been building up troops across from northern Gaza, preparing for an invasion. Tanks and troops moved in and out throughout the day and the military said it was carrying out "limited" operations to uncover explosives and tunnels.
A Hamas militant was killed and four were wounded in an Israeli airstrike in Beit Hanoun in northern Gaza just after midnight, Palestinians said. Israel said its air force targeted Palestinians planting a bomb near soldiers.
Early Tuesday, Israeli aircraft hit the student council building at the Islamic University in Gaza City, witnesses said, badly damaging it. No one was hurt. The university is a Hamas stronghold. The military said it hit a "compound used by terror groups for instructing and directing terrorists."
In the West Bank city of Ramallah early Tuesday, Israeli forces arrested three militants said to be involved in the abduction and killing of 18-year-old Israeli settler Eliahu Asheri last week. The military said a fourth suspect was arrested earlier. The kidnapping and killing compounded the tension over the captured soldier. Asheri's body was found Thursday near Ramallah.
After Shalit was seized in a June 25 raid on an army post, his captors demanded Israel free all imprisoned Palestinian women and minors in exchange for information about him. They later increased their demand to include the release of a further 1,000 prisoners.
The Palestinians apparently face a deadline of their own, reports CBS News correspondent Sheila MacVicar..
"Egyptian mediators, who have been key to the whole attempt to free the soldier, have told the Hamas leadership outside of Gaza that they have until this evening to respond, or all talks are off," MacVicar said.
In the week since the Israeli soldier was captured, Palestinians have come together and have come behind their government, sharing the view that something must be gained for all this pain, MacVicar reports. At the very least, Palestinians are hoping the captive soldier will be traded for the release of hundreds of Palestinian prisoners in Israeli jails. That is a demand that is almost universally popular, she says.
Early Monday morning, Hamas' military wing — one of the three groups holding him — issued a statement giving Israel until 6 a.m. Tuesday to "start" freeing the prisoners.
If Israel doesn't comply, "we will consider the soldier's case to be closed," the statement said, "and then the enemy must bear all the consequences of the future results."
Abu Obeida, spokesman for the Hamas military wing, later told The Associated Press that Israel must at least begin freeing the women and minors.
"Israel must understand that the resistance factions are serious in this matter. They will close this case if (Israel) doesn't deal with the demands," he said.
With the deadline less than eight hours away, Hamas sent out mixed messages late Monday. In Gaza, government spokesman Ghazi Hamad expressed hope for a diplomatic solution.
But Osama Hamdan, one of the most senior members of the exiled leadership, ruled out a compromise and threatened to abduct more Israelis.
"If this operation does not lead to the release of prisoners now, let's postpone talk ... and we will continue resistance. Other (Israelis) might be taken prisoner," Hamdan, Hamas' representative in Lebanon, told Al Manar television.
Hamas' options could be limited. Killing Shalit would remove their only leverage against Israel and would likely invite far harsher reprisals against Gaza.
"If God forbid, they should hurt the soldier, our operations will be far, far worse," Israeli Justice Minister Haim Ramon told Israel's Channel 2 television.
Israeli Prime Minister Ehud Olmert, who has repeatedly ruled out talks with the militants, said the government would not cave in to extortion.
"There will be no negotiations to release prisoners," his office said in a statement, adding that he holds the Hamas-led Palestinian Authority responsible for Shalit's safety.
But government and military officials said privately that Israel would pursue all options to get Shalit back. Israel has released prisoners before in lopsided exchanges for captured citizens or the dead bodies of soldiers killed in battle.
Mohammed Dahlan, a Palestinian legislator and close ally to moderate President Mahmoud Abbas, said the ultimatum was a negotiating tactic and that efforts to broker a compromise were continuing.
"What we care about now in the Gaza Strip is not to reach a point of no return," he said. "Everyone has an interest in getting out of this crisis."
The White House urged the militants to release Shalit.
"It is the responsibility of Hamas to return the Israeli soldier. That's how all this got started. We have also been encouraging Israel from the very beginning to practice restraint and continue to do so," White House press secretary Tony Snow said.
Many Palestinians say they do not wish Shalit to be harmed, but the demand to free some of the 9,000 Palestinian prisoners held by Israel is very popular here and it would be difficult for the militants to release Shalit without at least a token prisoner release.
"I think they should release the women and children and (the militants should) release the soldier. But without anything in return, they'll kill him," said Saked Abu Kosh, a 30-year-old pharmacist in the southern town of Rafah.
Egypt has been trying to mediate the crisis, but its efforts have been complicated by confusion over who has the authority on the Palestinian side to make a deal. The Hamas-led government says it had nothing to do with the abduction. The militants who seized Shalit are presumed to answer to the group's leaders in Syria, but those in Damascus say they bear no responsibility for the soldier.
Egyptian President Hosni Mubarak was in Saudi Arabia on Monday to coordinate with the Saudis on efforts to reach a deal over Shalit. Mubarak has already tried to enlist Syria's help.
In Turkey, Prime Minister Tayyip Erdogan said he had sent a special envoy to Syria to discuss ways of solving the crisis with President Bashar Assad. He did not say whether talks with Hamas' top leader, Khaled Mashaal, were scheduled.
Israeli Defense Minister Amir Peretz warned Damascus that he held it responsible for Shalit's fate because it harbored Hamas' leaders.
"I suggest that (Syrian President) Bashar Assad, who is trying to operate with his eyes shut tight, open his eyes, because he is responsible," Peretz said.
Dahlan spoke as the Palestinian parliament held its first session since Israel arrested dozens of top Hamas officials in the West Bank, including eight Cabinet ministers and 26 lawmakers, late last week.
"By arresting those lawmakers and ministers, Israel is trying to hijack the Palestinian political regime, but our people will protect our political regime," said parliament speaker Abdel Aziz Duaik.
Meanwhile, Israel continued its operations in Gaza, launching air strikes throughout the territory. Thousands of soldiers seized vacant areas near southern Gaza last week and Israeli tanks and troops took up positions outside northern Gaza early Monday and pounded the area with artillery.
Also Monday, Israel closed the Karni cargo crossing into Gaza, citing a security threat, just a day after Israel reopened Karni to allow badly needed humanitarian supplies to reach the coastal strip. The crossing is the main gateway for goods to enter Gaza.
In their statement Monday, Shalit's captors accused Israel of not "learning lessons" from the fate of other kidnapped soldiers. The last Israeli soldier kidnapped by Hamas, Nachshon Wachsman, died in 1994 in an Israeli commando raid on his captors' Jerusalem hideout.
Wachsman's mother, Esther, accused Israel's leaders of a lack of candor in dealing with hostage cases.
"I am not calling for the release of murderers, but (Israel's leaders) should not insult our intelligence because they have negotiated and they have given in to terror," she wrote in the Haaretz newspaper.