As first reported in The Times, Lord Mandelson will replace Dame Karen Pierce, whose term in Washington DC is due to end as Trump enters the White House in early 2025.
UK ambassadors are normally career diplomats or civil servants, but Downing Street said choosing a leading Labour politician "shows just how importantly we see our relationship with the Trump administration".
Labour Health Minister Stephen Kinnock said Lord Mandelson would make "an excellent appointment" as US ambassador.
Kinnock, whose father Neil Kinnock gave Lord Mandelson his first senior role in Labour in the 1980s, pointed to the former EU Trade Commissioner's "really strong experience in trade".
"He's got very good political contacts in Washington DC and I think his appointment would be a reflection of the importance of the US-UK special relationship and I think Peter Mandelson would be a very good person to take that relationship forward," Kinnock added.
But Lord Mandelson has been a divisive figures in British politics over many years.
He resigned twice as a minister - once for failing to declare a home loan from a cabinet colleague, and a second time over accusations of using his position to influence a passport application.
As a staunch critic of Brexit and advocate for global free-trade, he does not appear to be an obvious fit with the incoming Trump administration.
But the former cabinet minister and EU trade negotiator has enormous political experience, and Downing Street may have judged that sending someone so close to the big political figures in the UK may go down well at the White House.