Sen. John McCainJohn Sidney McCainMcConnell: GOP could try to repeal ObamaCare again after midterms Comey donates maximum amount to Democratic challenger in Virginia House race Live coverage: McSally clashes with Sinema in Arizona Senate debate MORE (R-Ariz.) praised Gov. Sarah Palin (R-Alaska), saying that he was proud of having her as his running mate and that he thought she could one day become president.

McCain, on Fox News on Monday, said that he was "very proud" of Palin, whom he picked to be the GOP vice presidential nominee. Asked by Fox's Sean Hannity if he spoke with Palin often, McCain said he did.

Hannity later asked McCain if he thought she could be president one day.

"Sure, sure," McCain replied. "But I'd also like to point out that it's very early in this whole season."

The Palin and McCain camps have been trading barbs since the latter stages of the campaign. McCain aides were reportedly frustrated that she veered off-message, that she spent more than $150,000 on clothes during the GOP convention and that she took a prank call on the radio.

Palin said last week that she would have been more independent from advisers if she could do the campaign over.

"If I were giving advice to myself back on the day my candidacy was announced, I'd say, Tell the campaign that you'll be callin' some of the shots," she said in Esquire. "Don't just assume that they know you well enough to make all your decisions for ya."

McCain said he wouldn't run again for president. He added that the party will have a "spirited" competition to decide its next presidential nominee, and that many governors aside from Palin could be in the running. He mentioned Govs. Bobby Jindal (La.), Jon Huntsman (Utah), Tim Pawlenty (Minn.) and Mark Sanford (S.C.) as leaders who will play an enhanced role in deciding the GOP's direction.

"One reason is because their talent, and another reason is because they govern," he said.