"Discussion is closed," said Abu al-Muthana, spokesman for the Islamic Army in the Gaza Strip.
"Whether he will be killed or not killed, we will not disclose any information about the fate of the soldier."
However, he also said that killing Corporal Gilad Shalit, captured in a cross border attack on June 25, would violate the captors' principles.
"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," al-Muthana said.
There was no immediate comment from the other two groups that were involved in capturing Shalit.
Hamas said on Tuesday that the three groups have pulled out of negotiations with Egyptian mediators trying to secure his release after Israel rejected a deadline, which expired at 6am local time, to release prisoners in exchange for Shalit.
"The militant groups withdrew their representative from the talks with the Egyptian mediators," Osama al-Muzaini, said on Tuesday.
Israel rejected any swap, even though the three factions, including the armed wing of the governing Hamas group, had said that it should respond by the deadline if it valued its soldier's life.
Avi Pazner, an Israeli government spokesman, said on Tuesday that Shalit was still alive.
"We know that until now Gilad Shalit is alive, we know that he is injured, that he was seen by a Palestinian doctor a few days ago," he told the French news channel LCI.
"We are sure of our information. As I speak to you, he is alive."
After Shalit's abduction, Israel mounted its first offensive in the Gaza Strip since quitting the territory last summer. It has massed troops and armour at the border.
Meanwhile, an overnight Israeli air strike in northern Gaza killed one militant and wounded two others. The army said they were planting bombs.
In Gaza City, an Israeli missile hit the Islamic University, a Hamas stronghold. The army said it had attacked a building used by Hamas to plan attacks.
"Woe betide them if they dare harm Gilad Shalit"
Haim Ramon, Israeli justice minister
Israel has also hinted that it may kill leaders of Hamas, whose government is under an international aid embargo, if Shalit is not freed.
Haim Ramon, the justice minister, said on Monday: "Woe betide them if they dare harm Gilad Shalit. The sky will fall upon them."
Hamas sources said Western diplomats, whom they did not name, had told the group that Israel had prepared a 13-man hit list, headed by Khaled Meshaal, Hamas's exiled leader and including Ismail Haniyeh, the Palestinian prime minister and Mahmoud al-Zahar, the foreign minister.
The Palestinian groups had first called on Israel to release 400 Palestinian women and youths from its prisons for information about Shalit.
Hamas's Izz el-Deen al-Qassam Brigades, the Popular Resistance Committees and the previously unknown Islamic Army subsequently demanded that Israel free 1,000 prisoners.
The factions demanded that
Israel free Palestinian prisoners
Unless the demands were met, the factions said, "the enemy will bear full responsibility for future consequences".
Mediation, led by Egypt, has failed to yield any results.
On Monday, the Palestinian United Nations observer urged the UN Security Council to compel Israel to end its offensive and free dozens of Hamas officials detained last week. He did not mention the capture of Shalit.
Western countries have urged the release of the soldier and called on Israel to show restraint.