Barack Obama is slowly starting the tight-lipped process of selecting a running mate and has tapped a party veteran to help, giving a signal that he believes his hard-fought Democratic primary race against Hillary Clinton is all but over.

Aides to the Democratic front-runner told FOX News the vice presidential vetting effort is only in its infancy. Officials said nobody has yet been tapped to spearhead the search, and that the campaign is just assembling a team that will formally begin the process once Obama secures the nomination.

Obama is still 61 delegates away from clinching the nomination, but has steadily focused his energy on presumptive GOP nominee John McCain.

Asked about his vice presidential selection process and criteria on Capitol Hill Thursday, Obama said: "No criteria right now... I'm going down to Florida. I still have to win the nomination."

Democratic officials said the Illinois senator has asked former Fannie Mae CEO Jim Johnson to begin reviewing potential vice presidential picks. Johnson did the same job for Democratic nominees John Kerry in 2004 and Walter Mondale in 1984.

The campaign tried to play down the development.

"News flash, Jim Johnson may help us look at a vice presidential process," a top Obama official said. "He's been around and that's not exactly a hot development. He's been doing this since Mondale. There's nothing official and all of the reporting about this is pretty rudimentary."

Officials said Johnson is just among those who will offer advice in the process. Former South Dakota Sen. Tom Daschle may also lend a hand in the process.

"Anyone can do that," the Obama official said. "This isn't a big deal."

Vice presidential searches are usually closely held secrets, but Obama campaign officials say the effort is being handled by a particularly tight circle of advisers.

The campaign did not want to discuss the effort because they are still engaged in a fading primary race against Clinton, with three contests left in Puerto Rico, South Dakota and Montana. The voting ends June 3. Obama has repeatedly declined to discuss possible running mates while the primary is ongoing.

"We're not commenting about this process," said Obama spokesman Bill Burton.

Possible options are Clinton; governors such as Arizona's Janet Napolitano, Kathleen Sebelius of Kansas and Tim Kaine of Virginia; foreign policy experts like former Georgia Sen. Sam Nunn, Connecticut Sen. Chris Dodd or Delaware Sen. Joe Biden; or other senators such as Missouri's Claire McCaskill and Virginia's Jim Webb.

Clinton spokesman Howard Wolfson said Thursday that "there have been no discussions with the Obama campaign about Senator Clinton being the V.P."

"It is clearly premature in the sense that he is not the nominee," Wolfson said of the running-mate search on a conference call with reporters. "But far be it from us to tell him when to begin a process. He presumably could have begun the process last year for all we know."

Time magazine reported Thursday that Bill Clinton is lobbying for his wife to be offered the No. 2 spot on the Democratic ticket.

He "is pushing real hard for this to happen," a source told Time.

Obama could also look outside the party to people such as war critic and Nebraska Republican Sen. Chuck Hagel or independent New York mayor Mike Bloomberg. Or he could look to one of his early prominent supporters like Daschle or try to bring on a Clinton supporter like Indiana's Evan Bayh.

They are meanwhile taking behind-the-scenes steps to move toward the general election campaign. Obama has 1,965 delegates to Clinton's 1,780, with 2,026 required to secure the party's nod under Democratic National Committee rules.

The Obama campaign is rapidly adding to its campaign staff, both at the headquarters and in general election swing states. Obama has been traveling to some of those battlegrounds -- Missouri, Michigan, Iowa and Florida in the last nine days -- while the campaign is registering voters across the country for the November vote. And top Obama organizer Paul Tewes is in discussions to take over the Democratic National Committee.

McCain has a head start and has been building his effort for several months since the GOP primary race wrapped up in early March.

McCain is hosting at least three Republicans mentioned as potential vice presidential running mates at his Sedona, Ariz., home this weekend -- Florida Gov. Charlie Crist, Louisiana Gov. Bobby Jindal and former Massachusetts Gov. Mitt Romney. A top aide said it's a social event with more than two dozen guests not meant for veep vetting.

Click here to read about how political oddsmakers view Obama's potential running mates.  

FOX News' Major Garrett and Aaron Bruns and The Associated Press contributed to this report.