Donald Trump mocked Michael Bloomberg, saying he "doesn’t have the magic," as the billionaire former New York mayor opened the door to a presidential run in 2020.
Mr Bloomberg, 77, took a major step towards declaring his candidacy for the Democrat nomination, indicating he would put his name on the ballot in Alabama. The state has the earliest registration deadline, at the end of this week.
A late entry by one of the world’s richest men could upend the Democrat race less than three months before the first voting takes place in Iowa on Feb 3.
Mr Bloomberg’s advisers said the media mogul was increasingly concerned none of the current Democrat candidates would beat Mr Trump.
However, Mr Trump said he would relish running against "Little Michael". Mr Bloomberg is 5ft 8ins tall.
The president said: "I’ve known Michael Bloomberg for a long time. He doesn’t have the magic to do well. Little Michael will fail.
"He’ll spend a lot of money. He’s got some really big issues, he’s got some personal problems, and he’s got a lot of other problems."
Mr Bloomberg was mayor of New York from 2002 to 2013. He has been a Republican, and an Independent, and registered as a Democrat last year.
He considered a White House run in 2016, but ultimately endorsed Hillary Clinton.
The former Wall Street banker, who founded a global media company, is worth at least $51 billion, according to Forbes.
He is the eighth richest person in America. The same list has Mr Trump at 275th, with an estimated fortune of $3.1 billion.
Mr Bloomberg has poured millions of dollars into gun control and climate change initiatives, and spent $100 million backing Democrat candidates in the 2018 mid-term elections.
His entry into the Democrat race would set up a showdown with Joe Biden for the support of the party’s moderate wing.
Mr Biden has suffered falling poll numbers and disappointing fundraising.
Mr Bloomberg has also been highly critical of Elizabeth Warren, currently the leading liberal candidate, over her plan for a wealth tax to help fund healthcare and college tuition.
On Twitter, Mrs Warren wrote "Welcome to the race, Mike Bloomberg!" and attached a link to her "Calculator for Billionaires," showing how much he would pay under her wealth tax.
Mrs Warren added: "It’s not enough just to have somebody come in, anybody, and say they’re going to buy this election."
Mr Bloomberg would face scrutiny of his three terms as New York mayor, including his defence of stop-and-search policies.
Some Democrats said it would be difficult for him to quickly build support across the country at this stage.
Harry Reid, the former Democrat leader in the Senate, said Mr Bloomberg called him on Thursday. Mr Reid said: "“It wasn’t just to wish me a good weekend. Bloomberg doesn’t do things halfway."
In a separate development, a new book by a senior Trump administration official, writing under the name ‘Anonymous’, painted a scathing portrait of the president.
The author alleged senior US officials considered a choreographed resignation dubbed a “midnight self-massacre” to raise the alarm over the Trump presidency.
It was claimed that Mr Trump once attempted a Hispanic accent while deriding women who cross the US-Mexico border claiming asylum.
He was quoted as saying: "’Oh, please help! My husband left me!’ They are useless. They don’t do anything for our country."
The author compared Mr Trump to "your elderly uncle running pantsless across the courtyard and cursing loudly about the cafeteria food" in a nursing home, and also questioned his mental abilities.
The White House dismissed the book as a work of "fiction".
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