However, he has not yet made a decision on whether to run, advisers add.
If Mr Bloomberg does indeed get on the ballot in Alabama, he will still have to register in other states which have later filing deadlines.
Some recent opinion polls have suggested that Ms Warren and Mr Sanders – who are regarded to the left of Mr Biden politically – might lose against Republican Mr Trump if either won the party’s nomination.
So far, three Republicans have said they will challenge Mr Trump in the party’s primary contest.
But it is seen as almost impossible that anyone will take the Republican mantle from the president.
What did Bloomberg’s adviser say?
In a statement, Howard Wolfson said: “We now need to finish the job and ensure that Trump is defeated.
“But Mike is increasingly concerned that the current field of candidates is not well positioned to do that.
“Based on his record of accomplishment, leadership and his ability to bring people together to drive change, Mike would be able to take the fight to Trump and win,” Mr Wolfson said.
President Trump, 73, has so far made no public comments on the issue.
The Democratic nominee for the presidential election on 3 November 2020 will be selected after a series of electoral contests across the country.
Candidates who want to get on the Democratic primary ballot in Alabama must submit the necessary paperwork by Friday.