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"I don't think (his selection is) out of the realm of possibility," said state Republican Party chairman Katon Dawson. He added "it's very exciting and a compliment to Lindsey Graham to be on the short list for vice president."
South Carolina's senior senator is considered to be very close to McCain, with the Associated Press reporting Graham probably travels more with McCain than anyone except his campaign staff and family.
"There's no doubt that they have a very good relationship and they both trust each other tremendously," said B.J. Boling, who was McCain's South Carolina spokesman during the GOP primary.
"I don't pretend to know all the considerations that go into picking a running mate, but I think loyalty, friendship and a similar world view are things you definitely consider," Boling said.
However, Graham said Friday a McCain-Graham ticket is out of the question. Graham is running for re-election to his Senate seat.
"It's an honor to be considered as somebody in the mix," he said. But, "I'm not on the list, I've never been on the list. I don't want to be on the list and if John asked me to be his running mate, I'd tell him to take a nap."
Graham also said he would be a bad strategic choice for the ticket in terms of the votes he could help gather.
Front-runners for the post include Minnesota Gov. Tim Pawlenty and former Massachusetts governor Mitt Romney, who both have roots in battleground states. One of those states is Michigan, where Romney, a former contender for the GOP nomination, was born. McCain is expected to win the Palmetto State.
Still, some believe the Arizona senator might be giving Graham another look.
There were reports this week that McCain also was considering Connecticut Sen. Joe Lieberman and former Secretary of Homeland Security Tom Ridge. But either choice might upset the GOP's conservative base because of both men's views on abortion. Talk radio host Rush Limbaugh predicted this week that "if (McCain) picks a pro-choice running mate, it's not going to be pretty."
Graham is anti-abortion.
University of South Carolina political science professor Blease Graham, who is not related to Lindsey Graham, said the South Carolina senator's visible role in the campaign may be spurring running-mate rumors.
"He's clearly a national opinion leader, but it's just where Graham thinks he will best serve the campaign and the state party."
Sen. Graham said he sees himself as most effective by staying in Congress. He said the trust he shares with McCain would make "me a valuable ally ... in the Senate."
He also denied he might take a position in a McCain administration. Graham, a former military lawyer, has been mentioned as a possible U.S. attorney general.
"Serving in a McCain administration as a Cabinet official is something I get asked about a lot, but I think my talents as an individual are best served in the Senate, both for the state and the nation," said Graham.
Graham faces nominal opposition in his November re-election bid. Myrtle Beach Democrat Bob Conley had $983, according to figures from the last reporting period, to spend on the race. Graham, on the other hand, had about $3.6 million in cash.
If Graham did become McCain's running mate or took a cabinet position, Gov. Mark Sanford would appoint a replacement who would serve for two years. There would be an election for the seat in 2010.
A McCain spokesman said the campaign doesn't discuss speculation about potential running mates.
