Senate Republicans pressed President Barack Obama at their Thursday lunch to stop bashing them on the stump, work with them more closely to craft a bipartisan budget deal and to approve the Keystone XL oil pipeline.

Obama committed to making a decision on the pipeline this year, and in turn pressed Republicans to give more on revenue, saying that without it, Democrats won’t support the entitlement reforms that he acknowledged would be needed to tackle the debt.

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.”

Obama set a tone of trying to find common ground, Republicans said.

“He said, ‘Hopefully I am a better president now than the day I started, all of us need to learn from our mistakes,’” Blunt said. “That was very much the tone; ‘I’m willing to do some hard things and I am willing to do it with you,’ but then there was always the ‘but here is what I have to have to be able to do hard things,’ which was more revenue.”

There did seem to be some areas of agreement, the Republican senators said, with Obama committing to exploring tightening cost-benefit analyses of new regulations and supporting revenue neutral reform of the corporate tax code, although Blunt said the president seemed to tie that to his demand for higher revenue elsewhere.

Indeed, on the broader budget stalemate that has dominated Washington, D.C., and the president’s meetings on the Hill this week with each party, there was only agreement that an agreement needs to be hammered out in the next few months — before next year’s midterm elections heat up.

Sens. John Hoeven of North Dakota and Lamar Alexander of Tennessee pressed the president on staying engaged with the Congress rather than going out on the stump.

Hoeven said he told the president, “You’ve got to grind it out with us until you get a deal,” instead of throwing up his hands and blaming Congress.

And Alexander gave Obama a bit of a history lesson, noting the previous cross-party relationships between presidents and members of Congress, from LBJ to Reagan. Alexander said Congress needs Obama to be “Moses,” and said 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.

Still, Hoeven and other Republicans pushed back against the president’s call for more statically scored revenue. (Some Republicans are open to new revenue coming from growth presumed from a simpler, more efficient tax code). They noted he got revenue in the fiscal cliff deal, and that Republicans have come to the table to back items such as means testing in Medicare, which effectively takes money from wealthier seniors.

Hoeven and Sen. John Barrasso of Wyoming both pressed the president on the Keystone XL oil pipeline. Amid the back and forth over the merits of the project, the president committed to making a decision this year, likely in a matter of months. Hoeven pushed for a faster decision, noting that the project has been studied for five years. While noncommittal, the president said that some of the claims by environmentalists against the pipeline were exaggerated, Hoeven reported.

In general, Republicans leaving the meeting appeared to be in good spirits, with Senate Minority Leader Mitch McConnell holding a news conference in the Ohio Clock Corridor with three GOP freshmen: Sens. Deb Fischer of Nebraska, Ted Cruz of Texas and Jeff Flake of Arizona.

Cruz said he asked Obama “if he agreed that the No. 1 priority for every elected official should be economic growth” and asked for answers on the issues of tax code reform and regulatory reform to reduce burdens on American business.

“I was encouraged by his answers to both questions,” the conservative firebrand said. The Texas Republican was quite complimentary of the president — a marked change of tone from the one he has used with Cabinet nominees — and said Obama seemed to support the GOP position on corporate taxes, “lowering rates” and “broadening the base.” Of course, not all Democrats agree with using the money generated by closing loopholes to lower rates, but the president’s conversation with Republicans seemed to indicate an openness to compromise.

McConnell seemed especially bullish on Obama’s tax talk, saying, “On the corporate side, it sounded like we had an agreement.”

The Kentucky Republican warned, however, that corporate tax policy needs to be reformed on the personal side as well, given the importance of s-corps, for example, but said he thought Thursday’s meeting was a good start.

Blunt said Obama seemed to tie corporate tax reform to his desire for revenue elsewhere.

Members cracked jokes as they were leaving the two-hour long lunch, as good a sign as any from a closed-door session that it went well.

In explaining his appreciation for Thursday’s visit, Flake said Obama’s outreach to the GOP is not unprecedented, noting that when he was in the House, the president invited 10 members to play basketball with him at the White House.

“We won and we were never invited back,” Flake said, to laughs.

On a serious note, Flake told reporters that on issues including the budget and immigration the president was clear he needs Republican 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 and 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.”