Prime Minister Ehud Olmert on Tuesday ignored a deadline to begin releasing Palestinian prisoners and instead issued a veiled threat against Syria, vowing to strike "those who sponsor" the militants in the Gaza Strip who seized a young Israeli soldier.

Olmert's comments signaled that a flurry of diplomacy throughout the region is still no closer to securing the release of Cpl. Gilad Shalit.

The Palestinian militants holding Shalit angrily said they would not release any more information about him after Israel ignored their ultimatum to begin releasing hundreds of prisoners early Tuesday. But they said they would not kill the soldier, after earlier signals that he would be harmed.

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CountryWatch: Israel

Speaking at a business conference in southern Israel, Olmert repeated his long-standing refusal to negotiate with Shalit's captors. "We won't negotiate with terror elements and we won't let anyone believe that kidnapping is a tool to bring Israel to its knees," he said.

He told the audience he has ordered the army to push forward with efforts "to strike terrorists and those who sent them and those who sponsor them," an apparent reference to Syria. "None of them will be immune."

Since Shalit was captured June 25 in a cross-border raid, Israel has made it clear that it holds Syria and Hamas' Syrian-based leader, Khaled Mashaal, responsible. Last week, Israeli warplanes buzzed the summer residence of Syrian President Bashar Assad. Israeli officials also have implied that Mashaal is a target for assassination.

Israel last week launched a ground operation in Gaza, along with a campaign of airstrikes, to pressure the ruling Hamas party to secure Shalit's release. Both Israeli defense officials and Palestinian security officials said the ground operation in northern Gaza was slowly expanding.

Hamas' military wing and two allied groups — the Popular Resistance Committees and the Army of Islam — have claimed responsibility for the abduction.

The militants holding Shalit had set a 6 a.m. (11 p.m. EDT Monday) deadline for Israel to begin releasing some 1,500 Palestinian prisoners in return for information about the soldier. The militants have given no details on Shalit's condition, although Israeli officials believe he suffered light wounds and is still alive.

After the deadline passed, a spokesman for the Army of Islam said Shalit's captors "have decided to freeze all contacts and close the files of this soldier."

"We will not give any information that will give the occupation good news or reassurance," said the spokesman, Abu Muthana. But, he added, "We will not kill the soldier, if he is still alive."

The groups had implied Monday that Shalit would be killed if their demands were not met.

Cabinet minister Roni Bar-On, who is close to Olmert, threatened harsh action if the soldier is harmed.

"It's safe to say ... the sky will fall on them if Gilad Shalit is harmed," he added. "If he is killed, we will react in ways the Palestinians haven't seen before."

Despite the tough public line, Israeli officials have privately said they would consider other options to get Shalit back. Israel has released prisoners before in lopsided exchanges for captured citizens or the bodies of soldiers killed in battle.

In Gaza City, Palestinian Prime Minister Ismail Haniyeh, a top Hamas official, called on Shalit's captors to protect the soldier and expressed hope for a peaceful resolution to the standoff.

"The government is exerting efforts with Palestinian, Arab and regional parties to end this case in the appropriate manner," Haniyeh said at the opening of a Cabinet meeting.

He said the government reiterates the need "to continue the political, diplomatic and negotiation efforts and not to close the door and use the language of wisdom and logic to end this."

Egypt has been leading international mediation efforts, including talking to Syria. Turkey, which neighbors Syria and is a key Muslim ally of Israel, also has sent an envoy to Damascus.

"We are dealing with more than one Middle Eastern government to see if the release of our serviceman can be secured. The fundamental principle is that his release must be unconditional," said Mark Regev, spokesman for Israel's Foreign Ministry.

In a new snag, Hamas lawmaker Salah Bardawil said the group had cut off talks with the Egyptians because of the lack of an Israeli response. "It's unreasonable to keep negotiating when there are no offers," he said.

"If there was a feeling there was an Israeli offer ... I think we cold reopen the door," he added.

Early Tuesday, Olmert and Defense Minister Amir Peretz decided to gradually step up the military operation in northern Gaza, defense officials said.

The army said the ground forces moved about a half-mile inside Gaza early Tuesday. Palestinian security officials said about 15 tanks had advanced slightly.

In an airstrike overnight, a Hamas militant was killed and four were wounded. Israel said its air force targeted Palestinians planting a bomb near soldiers' positions.

The Israeli offensive, which has left roughly half of Gaza residents without electricity, has raised concerns of a humanitarian disaster.

On Tuesday, Israel allowed Gaza's key cargo crossings to open to allow food, fuel and other key supplies into the area. However, the Erez crossing, used for people and vehicles to cross between Israel and Gaza, was closed due to a security alert, the army said.

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.