The Terminator versus the national security adviser for California governor in 2006?

Actor Arnold Schwarzenegger, weighing a run for the job, met Thursday with Karl Rove, President Bush's top political adviser.

If he decides to run, Schwarzenegger may face a challenge from national security adviser Condoleezza Rice, who has spoken to senior Republicans about running for California's top job, according to a Bush adviser. Rice's candidacy is a "real possibility," this adviser said, adding that Rice wants to take on an executive role.

A senior Republican official, who spoke on condition of anonymity, said Rice has told friends and associates she will not run for the Senate in 2004, but she has been careful in conversations not to rule out a gubernatorial bid two years later. This official, who talks frequently with Bush and his top advisers, said that while Rice does not seem to spend much, if any, time talking about politics, no one in the White House, including the president, would be surprised to see her run in 2006.

Rice's spokeswoman, Anna Perez, said the adviser "is not thinking about it, is not talking about it, is not considering it. She's very busy."

Rice, originally from Birmingham Ala., is on leave from Stanford University and was the school's provost from 1993 to 1999.

White House officials said Schwarzenegger simply dropped by to talk about an after-school program that California voters approved last year and to see what he could do to support U.S. troops overseas. Schwarzenegger also met with Lezlee Westine, director of the White House Office of Public Liaison; Education Secretary Rod Paige and lawmakers.

Politics didn't come up in the actor's session with Rove, the officials said.

But Rove plays a powerful role in selecting Republican candidates across the country and courting him could give Schwarzenegger's political fortunes a lift.

Schwarzenegger is also monitoring an effort under way in California to unseat Democratic Gov. Gray Davis through a recall campaign. "If it were to happen, he'd really have to think about it, because otherwise it could pass him by," said Rob Stutzman, communications consultant to the California GOP.

A political adviser to Schwarzenegger did not immediately return a call Thursday.