Movie star Arnold Schwarzenegger spoke privately with National Republican Senatorial Committee Chairman George Allen (Va.) during a visit to Capitol Hill last week, stoking speculation that he has not ruled out a run against Sen. Barbara Boxer (D-Calif.) in 2004.

Allen and Schwarzenegger spoke Thursday during a dinner hosted by the film celebrity, and then again privately, according to a knowledgeable Republican source.

George Gorton, Schwarzenegger’s political consultant, said his client only spoke with Allen at the dinner and has “no plans at this time to run for Senate.”

Schwarzenegger, who was in Washington to lobby for the reinstatement of federal funds for after-school programs, had a much more high-profile visit with White House political adviser Karl Rove on Thursday, presumably to discuss his interest in the 2006 gubernatorial race.

Republicans see Boxer as one of their top targets in the 2004 cycle, although they have yet to convince a strong candidate to enter the race.

“A lot of people are taking a serious look at the Senate race because Barbara Boxer is out of step with many voters in the state,” said National Republican Senatorial Committee Communications Director Dan Allen.

The potential Republican field runs from U.S. Treasurer Rosario Marin to Members of Congress — led by Rep. Doug Ose — to local elected officials such as Orange County Sheriff Mike Corona and finally to celebrity candidates, of which Schwarzenegger is clearly the best known and best thought of in GOP circles.

“I would welcome anyone into the race,” Boxer said when asked last week about a potential Schwarzenegger candidacy.

Schwarzenegger funded and served as the lead spokesman for the successful Proposition 49 campaign that won 57 percent of the vote in California last November. The proposition established state grants to pay for after-school programs.

Schwarzenegger has been openly considering a run for governor in 2006. But with the state wrestling with a budget deficit approaching $35 billion and a campaign to recall Gov. Gray Davis (D) under way, it’s possible that Schwarzenegger would take a look at a Senate run.

For either office, his vast personal wealth and nearly universal name recognition make him an attractive candidate for Golden State Republicans who have been decimated by losses at the state and federal levels over the past three cycles. The tremendous cost of running for statewide office makes it difficult for anyone not willing to personally fund their race or without access to a national fundraising network.

“The bottom line is that to be competitive in a California U.S. Senate race, you have to have $25 million,” said a Republican strategist intimately familiar with state politics.

“Anybody on the Republican side without a clear road map on how to get that $25 million or the ability to write a $25 million check will not be competitive,” the consultant added.

In Boxer’s 1998 Senate race against then-state Treasurer Matt Fong (R), she spent nearly $14 million to Fong’s $11 million.

In 2002, Davis spent a whopping $78 million on his successful re-election race against his Republican opponent, businessman Bill Simon, who doled out $36 million.

The most expensive California Senate race in history came in 1994 when Sen. Dianne Feinstein (D) was challenged by then-Rep. Michael Huffington (R). Huffington spent $30 million of his own money on the race, more than doubling the $14.4 million spent by Feinstein.

Feinstein prevailed, however, 47 percent to 45 percent.

For her part, Boxer pledged to stay her strategic course regardless of whom the Republicans nominate. “I do the same thing no matter who the opponent is,” she said.

Boxer’s primary focus at this point is raising the funds needed to finance her bid for a third term. After getting off to a slow start, Boxer picked up the fundraising pace significantly in the final six months of 2002, bringing in $878,000 with $1.4 million left on hand.

Boxer said that the advantage of running against Schwarzenegger is that “he has never done this before.”

“A lot of self-funders and celebrity candidates don’t know how hard it is to run in California,” she said.

Democratic Senatorial Campaign Committee spokesman Brad Woodhouse was more wry.

“There is only one thing to say about Senator Boxer’s prospects for returning to the Senate in 2005,” he said. “‘She’ll be baaack.’”