Dayot Upamecano had also been eyed before he agreed to leave RB Leipzig for Bayern Munich while Liverpool are making moves to sign his team-mate Ibrahima Konate, another young defender widely admired.
The Reds also have Ozan Kabak on loan from Schalke and it’s thought they will only sign one new defender, with Turk Kabak and Frenchman Konate the frontrunners.
White is a homegrown option who has taken the Premier League in his stride for Brighton, managing 28 appearances at centre-back and also in defensive midfield.
The 23-year-old has started all bar one of Brighton’s top-flight matches as Graham Potter’s Seagulls look set to avoid the drop back into the Championship.
The hopeful future England international impressed in the second division on loan at Marcelo Bielsa’s title-winning Leeds United last term and was named Young Player of the Year.
And his form for the Whites was closely monitored by United and Liverpool, with the centre-back still being watched by both clubs even now.
The Manchester Evening News say that White would be open to a move to Old Trafford at the end of the season and sees himself at a top club in the near future.
But Brighton do not want to lose their academy graduate and are expected to demand over £35million for him, having rejected multiple bids from Leeds last summer.
A price tag of around £50m has previously been mooted in some quarters, though that is still less than the fee Sevilla want for Kounde and the amount Real Madrid would demand for Varane.
White is likely to go head-to-head with Ole Gunnar Solskjaer’s United this weekend in the first round of matches after the international break, with the Red Devils hosting Brighton on Sunday night (7.30pm).
The MEN say White was added to United’s list of options back in December ahead of the 2021 January and summer transfer windows.
The ball-playing central defender could come in as a new partner for Harry Maguire with Eric Bailly still battling regular fitness problems and Victor Lindelof guilty of lapses of concentration and errors.
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Brighton boss Potter said of White, who signed a new four-year contract at the AMEX Stadium back in September, late last year that he is not concerned about transfer speculation.
“I have no worries about Ben getting his head turned,” he said back in early December.
“He is a level-headed guy who wants to play football and of course he wants to do it at the highest level, just like you want all those players to have those ambitions.
“He gets on with his work and he was totally professional in the summer despite the noise from outside the club, which you can’t control.
“He knows that, he’s a smart guy and we will carry on with our work and help him to enjoy his football, which is the most important thing. I have never been a coach that worries about losing a player.
“If you worry about losing players to teams higher up the hierarchy – because that’s ultimately what it is – and the club thinks selling the player is the right thing to do, then it’s a result of doing something well. I don’t think you should ever worry about that.
“It’s part of football and a lot of it is talk, speculation or nonsense. There’s always truth as well somewhere, but I am not worried about anything.
“I am just helping the players to improve, to get them to the highest level of performance that I can and then we will see.”