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 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.

And Tea Party favorite Sen. Ron Johnson of Wisconsin — who has proved a fiery foil to Obama’s agenda — said he heard from White House chief of staff Denis McDonough and now is willing to give “the president the benefit of the doubt.”

“If we’re going to solve these problems, it’s going to have to be done on a bipartisan basis ... and I think most Republicans are more than willing to work with this president,” he 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.