A report said the 78-year-old, famed for his use of polling and data, found a ticket of him and the former first lady and 2016 nominee would be a “formidable force”.
The Drudge Report, which famously was the first news organisation to reveal Bill Clinton’s affair with Monica Lewinsky more than two decades ago, said sources within the Bloomberg campaign reported he was even considering changing his official address from New York to Colorado or Florida, because the constitution said a candidate and their running mate ought not to live in the same state.
The report comes at a time of deep anxiety within the Democratic Party as to whether it has a candidate who can beat Donald Trump in November.
While both centrist Pete Buttigieg and progressive Bernie Sanders essentially shared the honours in the Iowa and New Hampshire primaries, the first states to vote, neither has established unchallenged frontrunner status.
At the same time, the campaign of former vice president Joe Biden, seems to be struggling, coming fourth in Iowa and fifth in New Hampshire.
Indeed, Mr Biden’s lacklustre campaign was one of the reasons Mr Bloomberg decided to enter the race, albeit very late. He has said he is prepared to spend billions of dollars to defeat Mr Trump.
Experts suggest there are few within the party who would wish to see a third presidential run from Ms Clinton, despite her recently telling British comedian and presenter Graham Norton, she had been “deluged” by requests.
Yet, her name as running mate might trigger less outcry, though it is questionable as to how many supporters of Mr Sanders would support such a ticket.
Last week, asked by Ellen DeGeneres if she had considered a possible vice presidential slot, Ms Clinton, 72, said: “Well, that’s not going to happen. But no, probably no.”
When she was pressed on the matter, she replied: “I never say never because I believe in serving my country, but it’s never going to happen.”
Mr Bloomberg’s campaign refused to rule out the possibility. In a statement sent to The Independent and other media, his communications director, Jason Schechter, said “We are focused on the primary and the debate, not VP speculation.”
Mike Fraioli, a veteran Democratic strategist said he was certain a woman would be one of the two names on the party’s ticket this year.
“One thing you can bank on, is that there is a woman on the ticket,” said Mr Fraioli. “Either as the candidate, or the vice-presidential candidate, there will have to be a woman on the ticket. Otherwise, we might as well pack up and go home.”
In addition to the two women still seeking the presidential nomination, Elizabeth Warren and Amy Klobuchar, among those mentioned as possible running mates should a man win the presidential nomination race, are senator Kamala Harris, and Stacey Abrams, who narrowly failed to win the Georgia governor’s race in 2018.
The report comes as polls showing Mr Bloomberg, who decided not to contest the early voting states of Iowa, New Hampshire, Nevada and South Carolina, soaring away from some of his rivals.
Much of that has come after spending $350m on television adverts in the handful of battleground states likely to determine the election.
An average of national polls collated by RealClearPolitics puts Mr Bloomberg in third place on 14 points, behind Mr Biden, 19, and Mr Sanders on 23.