Rep. Peter King (R) continues to show that he is seriously contemplating a run for Senate in 2010, this time by meeting with National Republican Senatorial Committee Chairman John Cornyn (Texas) on Capitol Hill on Monday to discuss the race, according to a senior GOP aide.

King has frequently floated his name for statewide office, but many Empire State political operatives in both parties believe he is closer to running this time than he has been in the past.

“He’s indicated that he’s very serious about the Senate seat,” the GOP aide said. “Sen. Cornyn thinks he would be a very strong candidate.”

King has seemed emboldened about a possible run since Caroline Kennedy (D) has emerged as the frontrunner to be appointed to the seat being vacated by Secretary of State-designate Hillary Rodham Clinton. Kennedy’s selection by Gov. David Paterson (D) isn’t guaranteed, however.

King has conceded that fundraising will be the major obstacle to mounting a statewide bid.