Barack Obama wasted no time formalizing his search for a running mate after clinching the Democratic nomination, naming a three-person team that includes Caroline Kennedy to spearhead the vetting process Wednesday.

Jim Johnson, a prominent Democrat and Obama financial backer who led the search efforts for former nominees Walter Mondale in 1984 and John Kerry in 2004, will lead the search team. Eric Holder, a former deputy attorney general and U.S. attorney for the District of Columbia under President Clinton, is the third member of the Obama team.

"Senator Obama is pleased to have three talented and dedicated individuals managing this rigorous process," Obama spokesman Bill Burton said. "He will work closely with them in the coming weeks but ultimately this will be his decision and his alone."

Obama advisers told FOX News the team will spend weeks screening potential nominees and operate somewhat separately from Obama's inner circle, to keep the process quiet and under the radar.

Obama aides said they want to focus national attention on the themes of his campaign against presumptive Republican nominee John McCain, and not on the hunt for a running mate.

Campaign officials said Kennedy, who is the daughter of the late President John F. Kennedy, and the two other team members have already begun compiling information on potential running mates. They disclosed no names.

But while Hillary Clinton has not conceded, her supporters are already talking up the possibility of her joining the ticket.

Bob Johnson, the founder of the Black Entertainment Network, was among them, and said he was doing so with her blessing.

Johnson said he had written the Congressional Black Caucus asking its members to urge Obama to place Clinton on the ticket.

Clinton "didn't direct me to do it but she certainly knows what I'm doing," said Johnson, who angered Obama's aides earlier in the race when he obliquely referred to Obama's confessed drug use as a young man.

Meanwhile, several previously uncommitted Democratic senators closed ranks behind Obama Wednesday, urging party unity.

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