Shalit's release has become a popular cause among Israelis, with many activists claiming the government has not done enough.
Olmert has tried hard to win the release of Shalit as part of his legacy, refusing to agree to an Egyptian-proposed ceasefire with Hamas in Gaza and opening the territory's border crossings more fully until Shalit has been freed.
Israel has suggested it could free hundreds of Palestinian prisoners in exchange for Shalit's release.
Mike Hanna, Al Jazeera's correspondent in Jerusalem, said: "As prime minister this is Olmert's last chance - from a timetable point of view - of doing something that he has based his prime ministership on, and that is bring Gilad Shalit back to Israel."
Meir Sheetrit, the interior minister in the outgoing government, said that the talks the envoys conduct in Egypt will be crucial and suggested that it might be more difficult for a deal to be reached with Netanyahu's administration.
"Today the issue will be decided for better or for worse. I hope it will be decided for the better," he said.
"Hamas understands that the days of this government are numbered.
"This government is ready to approve a deal, but if the government is replaced, there is no telling what will be."
Netanyahu, who has been in talks to form a coalition government, is hoping to present a unified administration to parliament in the next few days.
Officials from Likud have been re-examining the prospect of a coalition with Kadima, led by Tzipi Livni, the current foreign secretary who had previously dismissed Netanyahu's overtures and vowed to take her party into opposition.