WASHINGTON — Republican lawmakers said Sunday they welcome President Barack Obama’s courtship and suggested the fresh engagement between the White House and Congress might help yield solutions to the stubborn budget battle that puts Americans’ jobs at risk.

Yet the lawmakers cautioned that years of hurt feelings were unlikely to heal simply because Obama dined last week with Republican lawmakers. They also said they would not to rush too quickly into Obama’s embrace during three scheduled, and unusual, visits to Capitol Hill next week to win them over.

“He is moving in the right direction. I’m proud of him for doing it. I think it’s a great thing,” Sen. Tom Coburn, R-Okla., said. “I’m welcoming (him) with open arms. I think the president is tremendously sincere. I don’t think this is just a political change in tactic. I think he would actually like to solve the problems of this country.”

The White House charm offensive comes as automatic spending cuts have begun to take hold, and if Washington does not block them, they could cut jobs as varied as air traffic controllers, meat inspectors and Head Start teachers.

“I hope that this is sincere,” said Republican Rep. Paul Ryan, chairman of the House Budget Committee, who lunched with Obama at the White House last week. “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.”

His close friend, Rep. Cory Gardner, R-Colo., said lawmakers were unlikely to become fast friends with Obama after four years of being vilified in private and, in some cases, public.

“I hope that he’s genuine. But I don’t think we’re going to be doing the Harlem Shake any time soon together,” Gardner said.

Obama hopes this week to woo lawmakers to help avert a coming budget showdown — the next deadline is March 27, when the current short-term budget extension expires and a government shutdown looms.

Yet Congress is scheduled to leave town on March 22, meaning the president is working on a shortened timeline to avert the latest crisis. And the automatic spending cuts, known as sequester, remain in place despite both parties calling them ill-conceived and -executed.

Senate Democrats said they were ready to pass a spending measure to 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

“At the end of the day, we’re going to have to find a balanced solution,” said Sen. Tim Kaine, a Virginia Democrat whose state economy is closely tied to military contractors. The budget cuts are expected to be devastating to Virginia if they are not reversed.

Coburn said he was puzzled that it took so long for Obama — whom he called a friend after last week’s dinner — to make the effort to work across party lines.

“It shouldn’t be news that the president is reaching out in a bipartisan fashion,” Coburn said.

Obama seems to be making up for lost time after four years of frosty relationships with Capitol Hill. The White House said Obama planned to meet with the Senate Democratic Caucus on Tuesday, House Republicans on Wednesday and Senate Republicans and House Democrats on Thursday. Last week, Obama had Ryan and the Budget Committee’s top Democrat, Rep. Chris Van Hollen of Maryland, to lunch at the White House the day after he dined with a dozen Republican senators.

“I don’t know what he’s doing, but I would encourage him to do more of it,” said Republican former Florida Gov. Jeb Bush, a brother and son to U.S. presidents who visited five Sunday shows to keep buzz building toward a potential presidential candidacy in 2016. “People are more willing to do so if they believe that their partner is sincere in their efforts. And the only way you can do that is to engage them on a personal level.”

Republican Whip Kevin McCarthy, the No. 3 GOP lawmaker in the House, said Obama’s visit might be plain politics and tried to tamp down expectations.

“He has problems on both sides of the aisle. He hasn’t brought many people to the White House; he doesn’t know that many people in the House; he knows more people in the Senate because he served there for a couple of years,” McCarthy said. “He is coming to the House Republican Conference, which I think is positive. He has only done that once before. But he should come and listen, communicate and try to find where we can find common ground.”

Even members of the president’s own party said the outreach was needed.

“I think frustration by the American people — and frustration by members of Congress as well — have stemmed from the fact that a lot of these debates have taken place on TV or on the House floor but have not had very meaningful discussion,” said first-term Rep. Tulsi Gabbard, D-Hawaii.

Coburn, Gardner, Kaine and Gabbard appeared on NBC’s “Meet the Press.” Ryan spoke to “Fox News Sunday.” Bush appeared on CNN’s “State of the Union,” ABC’s “This Week,” “Meet the Press,” CBS’s “Face the Nation” and “Fox News Sunday.” McCarthy appeared on “State of the Union.”