Livni has so far made no comment since the release of the report from the government appointed Winograd report that heavily criticised the government’s handling of the war against Hezbollah last summer.
But hours before the extraordinary cabinet meeting, called to discuss the report's findings, the head of Kadima's parliamentary bloc in the Knesset called on Olmert to step down, saying it would be "suicidal" for the party, formed a year and a half ago, if he remained.
Olmert admitted that "there is no doubt that the report points at some extremely serious failures in the government's conduct and naturally first and foremost of me."
"A former cabinet minister said both Olmert and Amir Peretz, the defence minister, should resign"
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But he has refused to bow to the pressure to step down. His Kadima party cannot forcibly oust him from his post, as the party's charter does not spell out such a procedure.
He urged political rivals to think before using the report to advance their position.
"To all those who are in haste in order to take advantage of the report for political profit, I tell them not to be hasty," he said at the opening of the meeting.
But David Chater, Al Jazeera’s correspondent in Jerusalem, says the political momentum is still growing against the prime minister and a lot will depend on what Livni says after their meeting.
Chater said Livni is expected to give Olmert the ultimatum "either you quit or I resign."
Livni is one of the frontrunners to replace Olmert along with Shimon Peres, the deputy prime minister who has been prime minister on two previous occasions.
Another former prime minister, Binyamin Netanyahu, of the Likud party, a favourite in opinion polls, could also pose a challenge if Olmert's government is toppled.
He has not commented on the Winograd report so far but is expected to address a mass rally due to be held in Rabin square in Tel Aviv on Thursday.
The fallout against Olmert was exacerbated by the resignation of Eitan Cabel, a cabinet minister from the main governing partner Labour party on Tuesday.
"I can no longer sit in a government led by Ehud Olmert," Cabel said.