President Barack Obama has broken the ice with suspicious and distrustful Republicans on Capitol Hill during the past two weeks, but it hasn’t yet yielded any immediate breakthroughs on the budget stalemate that threatens to lead to yet another partisan debt showdown in a few months.

Obama’s meetings with the four congressional caucuses, and particularly with the GOP, were notable for just how rare they have been. And, at least on tone, he won plaudits in both chambers for engaging with Republicans instead of heading out to the stump and slamming them as defenders of the rich.

Those attacks proved successful in his November re-election but irritated many of the lawmakers he’ll need to craft and then push a bipartisan grand bargain across the finish line.

And while the White House has kept expectations low that his outreach will result in tangible benefits, aides say Obama will continue to engage after his trip to the Middle East next week and remains hopeful that the parties can ultimately come together on the budget and the rest of his agenda, including immigration and gun control measures.

Obama presented House and Senate Republicans with a choice — they can choose a deal that includes meaningful entitlement changes, including means-testing of Medicare and putting Social Security cost-of-living adjustments on a diet — or end up with no entitlement changes and no tax revisions. Sen. Roy Blunt of Missouri, the Republican Conference vice chairman, quoted Obama as saying, “I can’t provide the cover [for Democrats] to get entitlement reform done without revenue.”

But Republicans pushed back. That was particularly true in the House, where the leadership immediately went to the microphones to vow that they weren’t budging on their just-say-no position on new revenue.

And GOP leaders, including Speaker John A. Boehner of Ohio, continued to press the president to lead the country on the thorny issue of entitlements. Republicans want Obama to train his bully pulpit on the need for tough fiscal medicine rather than on attacking them.

One area where all sides seemed to agree — particularly in the Senate — is that the window is short before attention will shift to the midterm elections.

“I think a general sense on the part of Senate Republicans and the president both is that between now and the summer is a critical moment, and if we miss it, we miss the big opportunity maybe in the foreseeable future to get anything done,” Blunt said.

Sens. John Hoeven of North Dakota and Lamar Alexander of Tennessee also pressed the president to continue with the outreach, which has included calls to many Republican senators, a dinner with a dozen of them last week and this week’s three trips to the Capitol to meet with the Democratic and Republican caucuses.

“You’ve got to grind it out with us until you get a deal,” Hoeven said he told the president.

And Alexander gave Obama a bit of a history lesson, noting the previous cross-party relationships between presidents and members of Congress, from Lyndon Johnson to Ronald Reagan.

“I said look, this is historically the way presidents have gotten results. … This is the way to get things done. … Don’t just heckle and taunt us from the campaign trail.”

Alexander said Congress needs Obama to be “Moses” and added that if he puts forward a big deal that tackles entitlements and the debt, members in both parties would try to get it passed.

Hoeven predicted the odds of a grand bargain hammered out in the Senate then passing the House are “pretty good.”

“Because we need to do it,” he added hopefully.

Sen. Jeff Flake told reporters that on issues including the budget and immigration the president was clear that he needs GOP help. The Arizona Republican, who is a member of a bipartisan Senate negotiating group trying to hash out an immigration deal, said Obama opened his session Thursday by stating he’s not running for re-election anymore. The president said he would help move Democrats to the middle if Republicans could work on their conservative base.

“All of us got the sense that he realizes he has to reach out,” Flake said. “He’s going to need our help and we’re going to need his.”

Obama also engaged with Republicans in both chambers on the Keystone XL oil pipeline, saying he would decide this year, although he remained noncommittal. And he dangled a carrot to Republican senators of revenue-neutral corporate tax changes — though he still wants net revenue in other areas.

But repeatedly, Republicans, particularly in the House, seemed skeptical that the president would keep up his bipartisan push.

“Where he falls down is on the follow through,” said Rep. Joe Barton, R-Texas.

Rep. Mario Diaz-Balart, R-Fla., agreed.

“The question is, is there a follow-up?” Diaz-Balart said. “If there isn’t, then we’ll probably talk in another five years.”

Democrats, meanwhile, weren’t counting on any sudden conversion from the GOP.

“I hope it does good, but it’s certainly not going to do any harm,” said Senate Majority Leader Harry Reid of Nevada.

Meredith Shiner contributed to this report.