Senator Tom Coburn of Oklahoma had a scheduling conflict that evening, but he has been one of the few Republicans in Congress to have an open channel to the president, a relationship that began when Mr. Obama was a senator. Mr. Coburn, in an interview recently but before the president’s latest outreach, said that such legislative engagement was counter to Mr. Obama’s “personality type.”

“What he doesn’t realize is if he tried a different style, he’d get a whole lot more cooperation,” Mr. Coburn said, adding: “He’s really a neat guy. People don’t know that about him.”

But Mr. Obama insists that he tries.

“I promise you, we invite folks from Congress over here all the time,” Mr. Obama said at a White House news conference in January, when challenged about criticism of his infrequent outreach. He added, “Sometimes they don’t choose to come, and that has to do with the fact that I think they don’t consider the optics useful for them politically.”

Mr. Obama cited the example of Charlie Crist, the former Republican governor of Florida. In early 2009, Mr. Crist literally embraced the new president and his $800 billion two-year economic stimulus package when Mr. Obama visited Florida, thereby poisoning his candidacy for his party’s Senate nomination; it went to Marco Rubio, who then won the general election.

“It was the death knell for me as a Republican,” said Mr. Crist, who is now a Democrat.

Mr. Rubio, he recalled, put a photograph of the Crist-Obama embrace on his political literature and, at one rally, an angry Republican voter heckled, “Why don’t you just go hug Obama?” (These days it is Mr. Rubio, widely considered a potential 2016 presidential candidate, who has been trying to walk a hazardous line between getting legislative results — in his case a bipartisan immigration bill that Mr. Obama will sign — and not appearing too close to the president.)

More recently Gov. Chris Christie of New Jersey, another Republican seen as a presidential contender, enraged conservative activists with his warm reception of Mr. Obama as he toured the state after Hurricane Sandy, just before the November election.

Mr. Crist said his immediate thought was: “First Crist, now Christie. Look out buddy.” Before long came news of Mr. Christie’s snub by organizers of the annual Conservative Political Action Conference: He is not invited to this week’s conclave, which for years has served as a showplace for ambitious Republicans.