After a week of President Obama extending lunch and dinner invites to congressional Republicans, ahead of the next budget battle, party leaders are expressing cautious optimism about the president being sincere – not just leading a so-called charm offensive.

The invitations appear to be a change in political tactics for Obama, who since elected in 2008 has largely avoided courting Congress, preferring as of late to instead make his case to the public with campaign-style events.

He also plans to keep up the effort this week with three scheduled trips to Capitol Hill to visit Democrats and Republicans in both chambers, a likely effort to broker a “grand bargain” on tax and spending instead of another short-term fix.

"I hope that this is sincere," Republican Rep. Paul Ryan, chairman of the House Budget Committee, who had lunch Thursday with Obama at the White House, told “Fox News Sunday.” “We had a very good, frank exchange. But the proof will be in the coming weeks as to whether or not it's a real, sincere outreach to find common ground."

Obama’s apparent courtship follows $85 billion in spending cuts this year that kicked in March 1 and before the next deadline, March 27, when the current short-term budget extension expires and a government shutdown looms.

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Ryan, R-Wis., also told Fox his proposal for 2014 would balance the federal budget in 10 years, but is based in part on repeal of the president’s Affordable Care Act, which appears unlikely.

Though Republicans appeared optimistic, they also expressed concern that the president’s second-term efforts might be too late.

“I think he is sincere,” said Oklahoma Sen. Tom Coburn, who attended the hotel dinner with 11 other Republican senators Wednesday night.  

“But you know, you’ve got a lot of scabs and sores on people,” Coburn told NBC’s “Meet the Press.” “And it’s going to take a while for that to heal.”

The president will have to work quickly to achieve even a shorter-term deal, considering Congress is scheduled to leave town March 22.

Senate Democrats said they already have a spending measure ready for passage that would pay for day-to-day federal operations through September. The measure would impose automatic cuts of 5 percent to domestic agencies and 7.8 percent to the Pentagon.

Virginia Democratic Sen. Tim Kaine said Sunday that leaders of the Democrat-controlled Senate also are working on full budget for passage, the first in four years. And he expressed optimism about an eventual Hill deal.

“The two (chambers’) budgets will be different,” he told NBC’s “Meet the Press.” “We're going to have to find a balanced solution, and it will involve all elements. It will involve talking about revenues, talking about expenses, talking about entitlements. We have to do that.”

Still, some Republicans seem unconvinced about the president's recent efforts.  

"I hope that he's genuine,” Rep. Cory Gardner, R-Colo., told NBC. “But I don't think we're going to be doing the Harlem Shake any time soon together."

Republican Whip Kevin McCarthy question whether Obama’s efforts were about winning over the Republican-controlled House or “governing for all of America.”

“Only time will tell,” the California congressman told CNN’s “State of the Union.”

The Associated Press contributed to this story.