Former Mayor Edward I. Koch’s strategy seemed almost nutty at first: ask Jewish residents in Brooklyn and Queens to vote against an Israel-loving observant Jew for Congress to send a message to President Obama.

But the idea caught fire and appeared to be a significant factor in the special election on Tuesday in which Bob Turner, a Republican businessman, defeated Assemblyman David I. Weprin, a Democrat, in a race for the House of Representatives.

Now Mr. Koch is considering taking his critique of President Obama’s policies toward Israel to a national stage.

“I’m hopeful the president will read the tea leaves, will get the message – you’d have to be deaf not to,” he said, adding, “I’m hopeful that he will change his position.”

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.

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