Virginia Gov. Tim Kaine appears to be emerging as the frontrunner in the race to be named Barack Obama's Democratic vice presidential running mate.

The presumptive nominee has told close associates he has had "very serious" conversations with Obama and has provided documents to the campaign, several sources close to Kaine told The Washington Post.

Asked about the report on Tuesday, Kaine told WTOP Radio in Washington that he doesn't discuss his conversations with the campaign.

"You know, I save my advice, thoughts, etc. for those conversations and don’t really share them and I didn’t endorse Senator Obama to get anything  or to get appointed to anything," the governor said.

"It is flattering to be mentioned. My mom loves it she calls when she sees it, but that is for the campaign to decide and the campaign has made very very good decisions so far and I think they will make a very good decision, however they decide," he continued.

Sens. Joe Biden and Evan Bayh are also said to be on the short short list, according to the Post.

Obama was in Washington, D.C., on Monday to meet with his economic advisory panel, but also took time to visit the office of Eric Holder, who is leading the vice presidential vetting team with Caroline Kennedy. He is meeting with the House Democratic Caucus on Tuesday.

Kaine, a former Richmond mayor, would satisfy many of the criteria Obama is seeking in a running mate. He is an outside-the-Beltway type, suggesting he represents "change," Obama's key campaign theme. He also is very popular in an important potential swing state this year, and could attract Catholics, Hispanics and working-class voters.

However, Kaine has no foreign policy background to speak of nor has he achieved any major or notable accomplishment while governor.

Kaine and Obama have been friends for about three years, since Obama campaigned for Kaine during his gubernatorial bid. Kaine has since done the same for Obama during his presidential bid.

The Post reported that several people who have spoken to Kaine have discussed with him the depth of the vetting process, and one source told the paper that Kaine chief counsel Larry Roberts is coordinating with Obama's team.

Meanwhile, John McCain on Monday said he doesn't know who will be his vice presidential running mate and so many good candidates have emerged that he can't speculate when an announcement will come.

"Really, no," the presumptive Republican presidential nominee said when asked by CNN's Larry King whether he has made a decision about his No. 2.

McCain said he will make an announcement "just as soon as the process is completed" but said that process won't be driven by factors like the Olympics, which are scheduled to begin on Aug. 8, or the Democratic National Convention, which crowns Barack Obama as the party's nominee on Aug. 28 after three days of festivities.

"It wll be strictly on when we can arrive at a conclusion and honestly it is tough because we have so many highly qualified individuals," McCain said, adding that both men and women are being considered.

"If I start running down the list I will get in real serious trouble," he said.

Appearing on a remote telecast to supporters in Bermuda, McCain also told an audience at a Bakersfield , Calif., fundraiser that his vice presidential choice will be someone who "is not going to hurt the ticket."

"We are going through the process. There are so many highly qualified people in our party, and I won't mention names, ranging from people that have been stalwarts in our party for a long time to governors and senators and business people. It is a very tough decision," he said.

“You want to make sure you have a candidate that's not going to hurt the ticket. The second thing is, and I think it's the key criteria, is it someone who shares your principles, your values, your philosophy and your priorities. The hardest thing for the president is to establish priorities and I'm -- I think we're blessed with having a wealth of candidates. But I haven't -- we really aren't mentioning names," he said.

If not hurting the ticket is a goal, McCain may want to avoid Mitt Romney, who evangelical leaders have said would be a problem for McCain, according to The Washington Times. Though the former Massachusetts governor has strong support among economic conservatives, religious right leaders say social conservatives distrust his support for outlawing abortion and opposing same-sex marriage, the Times reported Tuesday.

Romney, who is also Mormon, has detractors among those who supported ordained Baptist minister and former Arkansas Gov. Mike Huckabee in the Republican presidential primaries, the newspaper reported.

"McCain and Romney would be like oil and water," evangelical novelist and Huckabee supporter Tim LaHaye was quoted telling the Times. "We aren't against Mormonism, but Romney is not a thoroughgoing evangelical and his flip-flopping on issues is understandable in a liberal state like Massachusetts, but our people won't understand that."

Huckabee was one of two potential running mates for McCain who appeared on FOX News on Monday night and offered varying degrees of enthusiasm about McCain's selection process.

Asked if he had to choose McCain's running mate, Huckabee said he would pick Minnesota Gov. Tim Pawlenty. The Republican convention is being held in Minneapolis-St. Paul during the week of Sept. 1.

Also on FOX News' Hannity & Colmes was Florida Gov. Charlie Crist, who refused to say whether he is being vetted for vice president, noting he can't talk about it.

"I can't discuss the process. I am enjoying being governor of Florida. It's the greatest job I've ever had in my life and I can't tell you how fun it is for me to wake up every day and have the opportunity to work for about 20 million fellow Floridians," Crist said, adding the job has not been offered.

Click here to read The Washington Post article on Virginia Gov. Tim Kaine.

Click here to read The Washington Times article on evangelical opposition to Mitt Romney.

Click here to see Florida Gov. Charlie Crist on FOX News.