But if Mr. Obama does not, Mr. Koch is weighing his options, he warned. Would he, for instance, go to Florida and talk to Jewish voters there?

“I might,” Mr. Koch said. “In fact, if it doesn’t work out that the president changes his position, I certainly will.”

The Obama administration already appears to be concerned about that possibility. About two weeks before the election, Mr. Koch said, the White House wanted to talk to him about his concerns, which stem in part from his belief that Mr. Obama is making harsher demands on Israel than the Palestinians in the effort to reach a Middle East peace agreement. Mr. Koch would not say who made the overture.

The administration asked him to explain “what it is that I felt aggrieved about,” he said, “which I did.” The conversation was friendly, Mr. Koch said. No one pressured him to stop criticizing the president, he said, nor was any particular agreement reached.

“It was, I think, ‘Let’s keep talking,’ ” Mr. Koch said.

Many factors affected the race for the Ninth Congressional District seat formerly occupied by Anthony D. Weiner, and several other prominent politicians ultimately endorsed Mr. Turner, including former Mayor Rudolph W. Giuliani and former Gov. George E. Pataki. But political observers said Mr. Koch’s role was crucial.