With the battle between President Obama and Mitt Romney now joined, attention has turned to possible Republican vice presidential picks.

New polls are offering likely GOP nominee Romney plenty of ideas -- from oft-mentioned names like New Jersey Gov. Chris Christie and Sen. Marco Rubio of Florida to some out-of-the-box choices such as Condoleezza Rice.

A Quinnipiac University survey out today has Christie, Rubio and House Budget Chairman Paul Ryan, R-Wis., leading the running-mate wish list. A third of independent voters, the bloc that often decides presidential elections, say Christie is a "good choice."

Rice, the former secretary of State, tops a poll released Wednesday by CNN/ORC International with support from 26% of Republicans and GOP-leaning independents.

Romney said this week that he's tapped longtime aide Beth Myers to conduct his search for a running mate, and hopes to name someone before the Republican National Convention in August.

Rice has emphatically said she has no intention of seeking elected office and doesn't want to be Romney's running mate. She would also pose a challenge for Romney, in that she is best known for her tenure in George W. Bush's administration and he remains unpopular.

Christie and Rubio have also been firm in saying they don't believe they'll be on the GOP ticket.

In both polls, name recognition appears to be a factor, which is why between now and the day Romney announces his choice Christie and Rubio will be repeatedly mentioned as possibilities.

Rick Santorum, Romney's chief rival for the GOP nomination until he dropped out last week, comes in behind Rice in the CNN/ORC poll. He's followed by Christie and Rubio.

In the CNN survey, Rubio is the favorite when supporters of the Tea Party movement are isolated in the VP findings. The freshman senator was elected in 2010 with support from the small government, anti-tax movement.