The eight-term lawmaker from Long Island said New York's GOP chairman Joseph Mondello "supports me 100 percent," and that state Conservative Party chair Michael Long has responded positively.
The son of an NYPD lieutenant, King said he "would genuinely represent the interests of blue-collar conservatives."
King's announcement is the latest twist in a political drama over who will succeed Clinton should she be confirmed as President-elect Barack Obama's secretary of state.
Her successor would be appointed by New York Gov. David Paterson, and about a dozen Democrats are vying for that position.
Kennedy's family has said she is interested in the seat.
Whoever is appointed would hold the job for two years, and run for election in 2010, against King or some other New York Republican.
King said the appointment of Kennedy would not scare him out of the campaign.
"Obviously it would be a challenge to run against Caroline Kennedy. She has the name identification and for all I know she's a wonderful person. But this is not an anti-Kennedy campaign," he said. "Nothing in life is easy. If anything, that makes the adrenaline pump a little harder."
King, 64, is the senior Republican on the House Homeland Security Committee.
"All my life I've been involved in New York issues and certainly since Sept. 11. I think I have as great a knowledge as anyone on homeland security and the threats to New York," he said.
He ran unsuccessfully for state attorney general in 1986, and has flirted with running for statewide office in recent years. He said his decision on whether to enter the Senate race will depend on how much money he can raise, since he estimates it will cost $35 million to $40 million to be competitive.
Oddly, the weakening of the New York GOP may give King more of an incentive to run statewide this time. The Democratically-controlled state legislature could re-draw congressional district lines in 2012 in such a way that makes it very difficult for King to keep his seat.
While King's conservative positions may hurt him in certain parts of the state, he has long been a friend of the state's politically powerful labor unions.
The Democrats mentioned as potential appointees are: Kennedy, New York Attorney General Andrew Cuomo, Buffalo Mayor Byron Brown, Nassau County Executive Tom Suozzi, teacher's union chief Randi Weingarten, as well as House members Carolyn Maloney, Kirsten Gillibrand, Steve Israel, Brian Higgins, Nydia Velazquez and Jerrold Nadler.
