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“Humpty Dumpty fell off and we’re going to put Humpty Dumpty back together, but it isn’t going to be done the same way that we were dealing with last week,” said Sen. Jon Kyl, Arizona Republican. “Some of the things that were kind of in play last week may not be quite the same now.”

His counterpart, Sen Edward M. Kennedy of Massachusetts, the Democrats’ chief negotiator in the grand bargain, said he saw good signs from statements over the weekend and said Mr. Bush has the chance to be convincing today.

“There’s nothing more persuasive than sincerity, and I think the president on this issue has demonstrated knowledge, awareness and sincerity, and I think that’s a powerful message,” he said.

Mr. Bush will likely find a polite but wary audience behind the closed doors. Many Senate Republicans think he is making a political and policy mistake by pressing for this immigration bill, and some accuse him of caving on key principles.

Sen. Jeff Sessions, Alabama Republican and an opponent of the bill, said Mr. Bush has even backed off of the principles he floated earlier this year in a Powerpoint presentation when the grand bargain was being formed.

“The principles that were stated in the Powerpoint presentation got watered down in every way,” he said. “The bill does not do what it needs to do. It is a political compromise.”

It will be difficult to find enough support among senators to break the filibuster.

In a test vote last week, 38 Republicans, joined by 11 Democrats and one independent, voted to filibuster the immigration bill. Just seven Republicans joined 37 Democrats and one independent in voting to limit the debate and force a final up-or-down vote.

Last week, Sen. Charles E. Schumer, New York Democrat, said he thought the vote on a final bill would be similar to the test vote. If so, that’s bad news for the bill’s backers, since the test vote garnered 50 votes in opposition — enough to kill the bill.