"He doesn't have convince me" too much," said Olmert at the start of his White House meeting with Mr. Bush.
Mr. Bush said he wants Abbas to lead the Palestinians "in a different direction."
The new situation in the Palestinian territories quickly became the main topic for a previously-scheduled meeting between the two leaders, as the two discussed how to shore up Palestinian President Mahmoud Abbas's shaky emergency government while isolating Hamas, reports CBS News White House correspondent Peter Maer.
Mr. Bush called Abbas "the president of all the Palestinians" and "a voice for moderation."
"I'm going to make every possible effort to cooperate with him," the prime minister said.
Meanwhile, hundreds of terrified Gazans fleeing Hamas rule were trapped at a main crossing between Gaza and Israel on Tuesday, hoping to gain permission to pass through Israeli territory to sanctuary in the West Bank.
In other developments in the Middle East:
Palestinian officials welcome the resumption of U.S. aid but admit that it won't solve their problems, reports CBS News correspondent Robert Berger.
"We have a major disaster, a major catastrophe, with this armed takeover in Gaza by these groups. And if we don't help ourselves as Palestinians nobody else will," said Palestinian legislator Saeb Erekat.
Presidents Bush and Olmert met at the White House in the aftermath of Palestinian turmoil that left Abbas, a Western-backed moderate, in control of one Palestinian government in the West Bank and his Islamist rival Hamas in control of the geographically-separate Gaza Strip.
"Our hope is that President Abbas and Prime Minister Salam Fayyyad — who's a good fellow — will be strengthened to the point where they can lead the Palestinians in a different direction," Mr. Bush said.
Olmert said he will be talking to Abbas but spoke of several prerequisites for progress towards peace.
They included a much more responsive Palestinian government and increased security efforts, Olmert said.
But Israeli analyst Dan Scheuftan says support of Abbas is a waste of time.
"Mahmoud Abbas is a hologram. He doesn't really exist. He is a rumor. His authority in the West Bank is nonexistent," Scheuftan said.