Republicans say they welcome President Barack Obama's courtship and suggest the fresh engagement between the White House and Congress may finally yield solutions to the stubborn budget battle that could cost Americans jobs.Yet the lawmakers cautioned Sunday that years of hurt feelings are unlikely to heal after their dinner last week with Obama. They say they will be careful not to rush too quickly into his embrace.

The White House has launched a charm offensive to woo Republicans. Obama dined with GOP officials last week, and he plans to visit members of both parties on three unusual trips to the Capitol this week.

U.S. Sen. Tom Coburn (R-OK) (Alex Wong/Getty Images)

Republican Sen. Tom Coburn of Oklahoma was at the Obama dinner last week. He tells NBC's "Meet the Press" that Obama is his friend.

(Copyright 2013 The Associated Press. All Rights Reserved)