WASHINGTON (CNN) -- Sen. John McCain said Sunday he hasn't seen signals Secretary of State Condoleezza Rice is interested in running as vice president.
A leading GOP strategist says Secretary of State Condoleezza Rice may want to run with Sen. John McCain.
His comments follow a report that Rice is angling for the No. 2 spot on the Republican presidential ticket.
"I missed those signals," McCain told reporters on his campaign plane. "I think she's a great American. I think there's very little that I can say that isn't anything but the utmost praise for a great American citizen, who served as a role model to so many millions of people in this country and around the world."
State Department spokesman Sean McCormack also denied Monday Rice was interested in the job.
"If she is actively seeking the vice presidency then she is the last one to know about it," he said. "She plans on going back West of the Mississippi to Stanford when she's completed her work as Secretary of State." Watch more of McCormack's comments »
The comments come after Dan Senor, a leading Republican strategist, suggested on ABC Sunday that Rice is mounting a behind-the-scenes campaign to be McCain's running mate.
"Condi Rice has been actively, actually in recent weeks, campaigning for this," Senor, a former Bush administration official said.
Senor cited Rice's recent appearance at the weekly meeting of Americans for Tax Reform -- a leading organization of Republican insiders -- as evidence she is attempting to cozy up to the conservative elite.
Speculation has long surrounded Rice as a potential vice-presidential nominee, especially after it became clear the Democratic presidential ticket will either feature a woman or an African-American for the first time in American history. Interactive: See who McCain may be considering »
But Rice has publicly said she has no interest in running for the job.
"I have always said that the one thing that I have not seen myself doing is running for elected office in the United States," Rice said in February. "I didn't even run for high school president. It's not in my genes."
But Gloria Borger, CNN's senior political analyst, noted Rice would instantly add star power to the Republican presidential ticket.
"Obviously, as an instantly recognizable national figure, Condi Rice would have to appear on any vice-presidential list," she said.
Choosing a Bush foreign policy official could prove tricky for McCain who has made an effort to distance himself from the administration's management of the Iraq war. Watch more of McCain's stance on the war »
Speaking with reporters Sunday, McCain said Rice bears some of the blame for mismanagement in Iraq.
"I think that pursuing the failed strategy and all who were involved in it bear some responsibility, he said. "Of course, I have put responsibility on the president, [former Secretary of Defense Donald] Rumsfeld, as well as Condoleezza Rice and to a lesser degree [Former Secretary of State Colin] Powell."
"I think it would be very difficult move for John McCain to make because it instantly fulfills exactly the argument that Democrats are presenting, that if you elect John McCain, what you're going to get is a third term of George W. Bush," Bill Schneider, CNN senior political analyst, said.
"What easier way to make that point than if you've got one of the architects of Bush's Iraq strategy on the ticket with him?" Watch more of Bill Schneider's analysis »
Another problem: Rice has virtually no relationship with the party's social conservatives -- a key voting bloc that is already wary of McCain. Watch more on McCain's relationship with the GOP base »
"One Republican strategist tells me she has 'done very little to court them,' and that could be a problem for McCain, who receives the same kind of criticism," Borger said.
But in a year where voters may be tempted to vote for a historic ticket, Rice has long been seen as a way to balance either the first African-American or female nominee of a major party.
"That's a consideration that you might want to take ... if you were anywhere along in the process," McCain said when asked if it would be a problem this year to put two white men on the ticket.
One named floated in GOP circles is former Massachusetts Gov. Mitt Romney -- a rival in the Republican primaries who recently joined McCain on some campaign events. Watch Romney discuss a potential McCain-Romney ticket »
"We're not in any point in the process, but I think that Americans overall want the most highly qualified person to serve anywhere in government," McCain said.
The comments come days after McCain acknowledged he had narrowed down his potential list of running mates to 20 names.
The Arizona Republican wouldn't hint at who was on the list but said he wanted the process wrapped up by the party's convention in September.
"I'd love to do it earlier in the run than later, but it depends on the process," he said. "We just really haven't gotten far enough along in the whole thing to really be able to even predict what we're doing," he said Thursday. E-mail to a friend