Richard W Painter, the chief White House ethics lawyer from 2005-2007, said Mr Trump would have “a massive fit” if the conservative news channel was bought from Rupert Murdoch’s Fox Corporation.
Mr Painter has joined a number of anti-Trump commentators, including the husband of senior Trump adviser Kellyanne Conway, in supporting the idea of Mr Bloomberg buying the company.
George Conway said on Friday that buying Fox News would be “the greatest thing [Bloomberg] could do for the country”.
Mr Bloomberg, who is the owner of the Bloomberg media company and a former mayor of New York City, has not officially announced a bid for the 2020 presidency but is thought to be seriously considering launching a campaign.
The billionaire businessman filed paperwork to enter the 2020 presidential race in Alabama on Friday, on the day of the deadline for submitting documents to be eligible for the state primary.
However, Mr Bloomberg has not campaigned in early-voting states, such as Iowa, New Hampshire, Nevada and South Carolina, and would need to break into a crowded field of Democratic candidates to challenge Mr Trump.
On Friday, the president appeared to welcome the idea of Mr Bloomberg running against him.
“He’s not going to do well, I think he’s going to hurt Biden actually,” Mr Trump told reporters at the White House.
“There’s nobody I’d rather run against than little Michael.”
Since 2016, Fox News has been accused of acting as a “mouthpiece” for the Trump administration, with the president often retweeting supportive video clips from the channel to defend himself.
Although Mr Trump recently criticised the channel for giving airtime to Democrats, he has frequently used Fox News clips in recent weeks to counter allegations of misconduct from the ongoing impeachment inquiry.
Fox Corporation, the parent company of Fox News, is not currently for sale and has a market capitalisation of $21.6bn.
Mr Bloomberg has a current estimated net worth of $52.4bn, according to CNN.
He has not made any indications that he would be willing or able to buy Fox News.