Washington ? As speculation swirls, Barack Obama and John McCain are knuckling down to the work of choosing running mates with their nominating conventions just weeks away.

Current and former governors and senators seem the most serious contenders, though most of those mentioned are playing coy about any discussions with either campaign.

“I’m just not going to talk about my conversations with the campaign,” Democratic Gov. Tim Kaine of Virginia said Tuesday as he declined to confirm media reports that he has provided financial documents to Obama for review. Another, Republican Gov. Tim Pawlenty of Minnesota, said he has decided to stop answering questions about a spot on McCain’s ticket because of all the gossip.

Among others believed to be getting close looks: Indiana Sen. Evan Bayh, Delaware Sen. Joe Biden and Kansas Gov. Kathleen Sebelius for Obama as well as former Massachusetts Gov. Mitt Romney and former Ohio Rep. Rob Portman for McCain.

Several more have been mentioned as well. But it’s appearing less likely that Obama – a Democratic Illinois senator – will choose former rival Sen. Hillary Rodham Clinton of New York. And it’s still possible that McCain – a Republican Arizona senator – will choose former Pennsylvania Gov. Tom Ridge, whose support for abortion rights might upset conservatives whose enthusiasm McCain needs.

This much is certain: Obama and McCain have been regularly huddling behind closed doors with a small circle of advisers to examine the backgrounds and records – and weigh the political implications – of at least a handful of prospects.

Each nominee-to-be may even be holding private one-on-one meetings with vice presidential hopefuls or polling possible tickets to see how they would fare in certain target states. It’s tough to tell; such details are tightly held.

At the same time, possible picks are trying out for the part in public. They do numerous TV interviews on behalf of the candidates and campaign alongside them, almost certainly at the behest of campaign advisers who want to assess how each handles the media and campaign rigors.

Obama could thoroughly review his options during his planned Hawaiian vacation and then name his choice in the week before the Democratic convention in Denver at the end of August. McCain is seriously considering naming his No. 2 in the few days separating the two conventions in the hope of stunting any post-convention surge in the polls for Obama.

Recently, each has left the impression his search has reached an advanced stage.

Obama met Tuesday with top aides and his search committee in Washington for the second time in as many days, and at least the third time this month.

He told NBC’s Tom Brokaw on Sunday: “I’m going to want somebody with integrity; I’m going to want somebody with independence, who is willing to tell me where he thinks or she thinks I’m wrong; and I’m going to want somebody who shares a vision of the country – where we need to go. That we’ve got to fundamentally change not only our policies but how our politics works; how business is done in Washington.”

That comment seemed to cast doubt that Obama would choose Clinton, who has spent more than a decade in Washington as a first lady and New York senator. During the primaries, Obama portrayed Clinton as the ultimate Washington insider while suggesting he offered a fresh approach above partisanship.

Even so, Obama told Brokaw: “I’ve said consistently that I think Hillary Clinton would be on anybody’s short list. She is one of the most effective, intelligent, courageous leaders that we have in the Democratic Party.”

McCain, for his part, didn’t give away much more Monday.

He told CNN’s Larry King that he’s considering both men and women and, “I will announce it just as soon as the process is completed.”

Aides to both candidates refuse to describe the selection process and bristle when asked directly about one person or another; neither side appreciates prospective running mates who promote themselves.

Even now, it’s possible long-shots could be chosen, like Republican Sen. Chuck Hagel of Nebraska for Obama, Alaska Gov. Sarah Palin for McCain, or New York Mayor Michael Bloomberg, an independent, for either.

And names that were hot just months ago seem to have grown cold, like Florida Gov. Charlie Crist, Louisiana Gov. Bobby Jindal and South Dakota Sen. John Thune among Republicans, and Democrats like Ohio Gov. Ted Strickland and Virginia Sen. Jim Webb have taken themselves out of the running.

So far, the few who know for certain the names remaining on each short list aren’t talking.