Within the governor's large circle of political advisors, there is no consensus about how to approach the convention. Some believe it is a mistake for him to go at all; others see in the convention an irresistible forum for what could be a defining speech.
Schwarzenegger "is aware it could possibly subtract from his ability to accomplish much of what he wants to in California, because the president is not nearly as popular here as he is," said one political consultant, speaking on condition of anonymity. A Los Angeles Times poll in April found that 54% of California voters disapproved of Bush's job performance.
Another friend and advisor said he warned Schwarzenegger privately that he should refuse even the title of honorary chair of Bush's California campaign. The governor accepted the title anyway.
Schwarzenegger's "numbers are through the roof," this advisor said, "and Bush is a catalyst for anger among Democrats.... I just think it's a bad mistake" to attend the convention.
Still, for both the Bush campaign and Schwarzenegger, a convention speech could pay off. It would be Schwarzenegger's national debut of sorts as an elected official. Whether he gets a prominent speaking slot or not, he is assured that an eager national media will offer a platform where he can tout his California agenda.
"Back East he's still the celebrity action hero. This is an opportunity to show the rest of the country that he's not just an action hero but a serious and effective governor," said Dan Schnur, a Republican political consultant close to aides in the Schwarzenegger administration.
For the Bush campaign, Schwarzenegger's appearance might send a reassuring signal to moderate or undecided voters that the Republican Party is open to dissonant voices not necessarily in step with the conservative wing. Schwarzenegger supports abortion rights, opposes a constitutional amendment banning gay marriage and supports stem cell research.
"Whereas the convention hall will be dominated by conservatives -- as it always has been -- communicating directly with undecided voters in key, targeted states will be a mission of this convention. Arnold Schwarzenegger is someone who can do that," said Scott Reed, a Republican strategist who managed Bob Dole's 1996 presidential campaign.
There is another potential bonus for convention planners. In an era where there is little drama at political conventions, prompting networks to tune out, a politician with Schwarzenegger's celebrity could serve as a lure.
"He's a fascinating political actor and I think he'll rouse a lot of public interest and curiosity. And I think how he's working in California is a great story and how his views in some ways differ and in some ways don't differ from the mainstream Republican Party is a great story," said Neal Shapiro, president of NBC News.
Still, the convention could prove dicey for Schwarzenegger if it grows more partisan. But there's a solution for that:
"I imagine Schwarzenegger and his team will accept an offer to speak at the convention and get in and get out," Reed said.