"We are sure of our information. As I speak to you, he is alive," Israeli cabinet spokesman Avi Pazner told French television.
Israel flatly rejected the captors' 6:00 am deadline to free prisoners or face the consequences.
After the deadline, one of the Palestinian militant groups that abducted the 19-year-old soldier said no further information would be given on the captive's fate.
"Discussion is closed," said Abu al-Muthana, spokesman for the Islamic Army in the Gaza Strip, told Reuters.
"Whether he will be killed or not killed, we will not disclose any information about the fate of the soldier."
But Abu Muthanna later told AFP the captors did not want to kill the serviceman.
"Some people thought that the groups that carried out the operation will kill him but our Islamic values tell us that prisoners should be respected and not killed."
There was no immediate comment from the other two groups that were involved in capturing Corporal Gilad Shalit on June 25.
The Islamic Army is the least well known of the groups. The others are the armed wing of the governing Hamas Islamist group and the Popular Resistance Committees.
Abu al-Muthana said he expected a joint statement later today.
There was no comment from the Israeli military, which rejected the ultimatum and militant demands for the release of 1,000 prisoners as well as Palestinian women and minors held in Israel.
Israel has waged a massive assault on the Gaza Strip, returning troops and tanks to the territory for the first time since leaving last year after a 38-year presence.
"We will not conduct any negotiations on the release of prisoners," Prime Minister Ehud Olmert's office said, holding the Palestinian Authority responsible for the safety of Corporal Gilad Shalit.
Olmert then gave the green light to keep up military operations, after consulting defence officials, while Justice Minister Haim Ramon warned of even stronger repercussions.
"If, God forbid, they should hurt the soldier, our operations will be far, far worse," Mr Ramon told Israeli television.
As Israeli officials warned the crisis could last months, the Hamas-led cabinet appealed to the captors to spare Shalit's life and expressed hope a political solution could still be found.
"We believe there remains a chance to reach an acceptable formula," said Ghazi Hamad, the spokesman of the Islamist-led government.
Saudi King Abdullah and President Hosni Mubarak, who has tried to mediate, "condemned the current Israeli escalation, and agreed that mediation should continue until a solution is found that would put an end to the escalation".
Syria, which has come under fire from Israel and Washington for sheltering exiled Hamas leader Khaled Meshaal, also condemned "Israel's aggressive stand and its unjust accusations against (Palestinian) national forces".