Sen. Bob Bennett (R-Utah) is not ruling out an independent run for his Senate seat.

Bennett was defeated on Saturday at the Utah Republican Convention, finishing third to rivals Mike Lee and Tim Bridgewater, who will fight for the Republican nomination in a June 22nd runoff.

Bennett has the option to run as a write-in candidate, and suggested Tuesday he wasn't closing the door on any of his choices.

"We've made the firm decision not to make any decisions as to what I do from now on until we have a little time to think about it," he told reporters.

But he can't count on the Republican Party to help him.

National Republican Senatorial Committee Chairman John Cornyn (Texas) said the NRSC won't support a write-in effort by Bennett.

Cornyn said the NRSC would support instead whichever nominee is selected by Utah's Republican voters.

Cornyn said as a rule, the NRSC doesn't support independent bids.

"We'll support the Republican nominee," Cornyn said. "That's what we do. We're not going to support any independent candidates."

When asked about Bennett, Cornyn slowly shook his head no.

"He's got a very stellar record, one he should be very proud of," Cornyn said. "These are not lifetime jobs or entitlements. They're subject to the revocation of the voters, and the voters are upset with what's happening in Washington and they've spoken. We'd all do well to take heed."

The NRSC criticized Florida Gov. Charlie Crist (R) when he decided to make an independent bid for the Senate.