President Obama shakes hands with House Speaker John Boehner as they participate in the unveiling of a statue in honor of civil rights activist Rosa Parks, in Statuary Hall in the U.S. Capitol on Feb. 27. (JASON REED/REUTERS)

Two weeks after launching a high-profile charm offensive targeting Capitol Hill, President Obama and his aides have taken their effort behind the scenes — quietly pushing for cooperation between the White House and congressional Republicans on key disputes.

Before departing for Israel last Tuesday, for example, Obama called Republican Sens. John McCain (Ariz.) and Lindsey O. Graham (S.C.) to discuss immigration reform and other issues. The White House legislative affairs office reached out to Rep. Tom Cole (R-Okla.) last week after he spoke of being ignored. And Obama counselor Pete Rouse worked with Sen. Lisa Murkowski (R-Alaska) on resolving the impasse over Interior Secretary-designate Sally Jewell’s nomination.

Lawmakers and aides say the effort has begun to yield modest dividends. Last week, Congress managed to pass a continuing resolution averting another potential government shutdown.

“It’s sort of like the two sides are looking across the table and thinking, ‘We really are going to have to live in this house for the next four years. Let’s divide up who does the dishes: I’ll take Tuesday, Thursday and Saturday,’ ” said Cole, who has broken ranks with his party on occasion. “I sort of see the CR as a confidence builder.”

But diplomacy still has its limits. On Friday, the White House formally withdrew the nomination of Caitlin Halligan to join the U.S. Court of Appeals for the District of Columbia Circuit after Republicans had blocked her appointment for years.

House Budget Committee Chairman Paul Ryan (R-Wis.) had lunch with both Obama and his panel’s ranking Democratic member, Rep. Chris Van Hollen (Md.), on March 7. But last week, Ryan’s aides had no contact with White House officials as they pushed through their conservative budget plan.

By contrast, Obama called Sen. Patty Murray (D-Wash.) to thank her for her work on the Senate Democrats’ budget blueprint.

“This is not a week where they are going to pick up the phone and call House Republicans,” Van Hollen said, referring to White House officials.

“After a week of constructive meetings, the president and his senior team have continued conversations with members,” White House spokeswoman Amy Brundage wrote in an e-mail, adding: “That outreach will continue.”

Van Hollen said that one of the inherent challenges is that House GOP leaders are willing to engage with Obama only in a public context — such as during a budget conference — for fear of alienating their rank-and-file members.

Rep. Scott Rigell (R-Va.), for example, said he hesitated for a moment when the White House called him Feb. 25 as he was sitting in Norfolk’s Piccadilly Cafe to ask whether he wanted to travel to Hampton Roads with the president aboard Air Force One.

“I had to do two hours of thinking in about four of five seconds,” Rigell said, knowing he would come under attack from conservatives for having a private talk with the president. “I just went, ‘Oh my, here we go.’ But it was a good conversation, I’m glad we had it.”

Rigell, who said he told Obama it was urgent to cut federal spending, said the discussion did not transform his relationship with the White House.

“I’d love to say we’re text messaging back and forth. That is not happening,” he said, adding he hasn’t heard from the administration since. “I do feel if I needed to reach out, I could do that.”

Sen. Ronald H. Johnson (R-Wis.), who had dinner with the president and 11 other Republicans at the Jefferson Hotel on March 6, said Obama made a point of telling the group that Americans don’t realize they get $3 back for every dollar they pay into the federal entitlement system. But the president needs to do more than acknowledge it privately, Johnson said: “He needs to start honestly telling the American people how serious the problem is.”

Ryan described the charm offensive as “helpful” during an interview on Greta Van Susteren’s “On the Record” on Thursday, adding: “The question is, is it real and will it last? If it is real and if it does last, then I think we’ve got a chance of getting a down payment on the problem fixed.”

For the most part, these conversations are taking place outside congressional leadership circles.

Cole told The Washington Post earlier this month that he hadn’t been contacted in months by the White House legislative affairs office. Staffers quickly reached out to schedule an appointment between Cole and the White House’s new legislative director, Miguel Rodriguez.

“They’ve been trying,” Cole said, saying members appreciated that Obama spent an hour and a half speaking to the House GOP Conference of March 13 and shook hands and took photos with any member who came up to him afterward. “He was not whisked out. That was not lost on anybody.”

House Administration Committee Chairman Candice S. Miller (R-Mich.), who has clashed with the administration over canceled White House tours and other issues, recently huddled in her office with Rodriguez on how to promote natural-gas-powered vehicles.

“If you’re looking for a Republican to work with you on that, you’ve found one,” she recalled telling Rodriguez.

Miller struck a similarly positive tone when she got a chance to ask the president a question during his recent closed session with House Republicans, who face regular attacks from the Democratic Congressional Campaign Committee.

“Please don’t be a stranger,” Miller remembered telling Obama. “And please don’t believe anything the DCCC puts out on the House Republicans.”

Rosalind S. Helderman contributed to this report.

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