Israel's Channel 10 TV reported a leaflet was circulating in the Palestinian areas in which the militants holding the soldier said they would issue no further information about the soldier after Israel ignored the deadline. The report could not immediately be confirmed.
A day earlier, the militants indicated they might kill the captured soldier if their demands were not met. Calls after the deadline expired to Hamas leaders and to Abu Obeida, spokesman for the Hamas military wing, were not answered.
There was also no word on the condition of the soldier, Cpl. Gilad Shalit, 19. Though Israel rejected negotiations or prisoner releases and Hamas insisted Israel would have to pay a price for the soldier's return, there were first signs of erosion of the hard lines on both sides.
If Israel hoped the offensive would drive a wedge between the Palestinian people and the militants, they were wrong, reports CBS News correspondent Sheila MacVicar. She reports that the cause of freeing the thousands of Palestinian prisoners in Israeli jails is a hugely popular cause in Gaza and that people believe that the militants will be able to trade Gilad for their freedom.
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 freedom for the soldier.
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 Hamas 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.
In the West Bank city of Ramallah early Tuesday, after surrounding a Palestinian police building, Israeli forces arrested three militants said to be involved in the abduction and killing of an 18-year-old Israeli settler last week. The military said a fourth was arrested earlier.
The kidnapping of the teen, Eliahu Asheri, added tension to the situation surrounding the capture of the soldier. Asheri's body was found Thursday near Ramallah.
The soldier was taken during a Hamas-directed cross-border raid June 25 on an Israeli army post at a Gaza crossing point. Two other soldiers were killed in the attack.
The militants holding the soldier — Hamas and two small groups affiliated with Hamas — announced their ultimatum early Monday on a Hamas-sponsored Internet Web site, giving Israel 24 hours to begin releasing prisoners, implying that they would kill the captive if their demands were not met.
Since the June 25 raid, however, there has been no word of Shalit's condition. The Israeli military said he was wounded in the Palestinian attack and was presumed to be alive.
In the hours before the deadline expired, Israel kept up its pressure on Hamas with airstrikes.
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."
The military said it also hit another Hamas facility in northern Gaza.
Also, Palestinian residents said several Israeli tanks and a bulldozer entered the northern Gaza town of Beit Hanoun after midnight. The military said they were still outside. Israel has been building up troops across from northern Gaza, preparing for an invasion.
A Hamas militant was killed and four 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' positions.
Reacting to the Tuesday morning deadline, 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 they back the demand to free some of the 9,000 Palestinian prisoners held by Israel.
"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.
In Turkey, Prime Minister Tayyip Erdogan said he sent an envoy to Syria to discuss ways of solving the crisis with President Bashar Assad.
Also Monday, Israel closed the Karni cargo crossing into Gaza, citing a security threat, just a day after Israel reopened it to allow basic food products to reach the coastal strip. The crossing is the main gateway for goods to enter Gaza.
In the West Bank town of Jenin early Tuesday, Israeli soldiers shot and killed an 18-year-old Palestinian who was planting a bomb, Palestinians said. The military said soldiers returned fire at armed Palestinians.