As a result, the paper reported some aides are now telling Bush that if he wants to forestall more defections, it would be wiser to announce plans for a far more narrowly defined mission for US troops that would allow for a staged pullback.
The president had originally rejected this strategy in December when it was proposed by the bipartisan Iraq Study Group.
"When you count up the votes that we've lost and the votes we're likely to lose over the next few weeks, it looks pretty grim," the New York Times quotes one senior official as saying.
In a sign of growing concern in the administration, Robert Gates, the US defence secretary, cancelled his Latin American tour on Sunday to attend meetings on Iraq.
Last week, Stephen Hadley, US national security advisor, was called in from a brief holiday to join discussions on Iraq, which included Karl Rove, a US political strategist and Joshua Bolten, the White House chief of staff, according to the report.
Officials describe Hadley as deeply concerned that the loss of Republicans could accelerate this week, a fear shared by Rove, the report said.
But they also said that Rove had warned that if Bush went too far in announcing a redeployment, the result could include a further cascade of defections - and the passage of legislation that would force a withdrawal by a specific date, the paper said.
"Everyone's particularly worried about what happens when McCain [John, a Republican senator] gets back from Iraq," one official is quoted as saying.
McCain has been a strong supporter of the "surge" strategy, but is facing political troubles in the race for the Republican nomination for president.
McCain's poor performance in presidential nomination polls, attributed to his position on Iraq, has fuelled speculation that he may declare that the Iraqi government is incapable of reaching the kind of political accommodations that Washington considers necessary for overall success, the New York Times said.