The England international is now understood to be the Red Devils' leading centre-back target for the 2024-25 season, and they have been hopeful of capitalising on Everton's financial situation in hopes of a cut-price deal.
However, the Toffees do not have such a great need to accept a cut-price sum after staving off relegation from the Premier League, and they are believed to be holding out for a £70m sum to let him leave.
Everton apparently feel that Branthwaite commands the same extortionate transfer fee as Harry Maguire, while Man United tested the waters with an opening bid of £35m rising to £43m with add-ons.
The Toffees quickly laughed off the Red Devils' first approach and are holding out for at least double that amount to enter talks, but according to The Mirror, they have been told to "get realistic" by their Mancunian counterparts.
Man United 'have just £50m' to spend
The report adds that Man United are not backing down on their initial valuation, which they believe is fair, and they have told Everton to consider Branthwaite's real market value if they want to shake hands on an agreement.
As things stand, the Red Devils supposedly only have a transfer budget of around £50m to work with this summer, as they are one of many limited by the constraints of Financial Fair Play.
However, player sales could help bump up that figure significantly, and only seven of Erik ten Hag's most important troops are thought to be completely safe from the chop this summer.
All of Alejandro Garnacho, Rasmus Hojlund, Kobbie Mainoo, Andre Onana, Diogo Dalot, Lisandro Martinez and Amad Diallo are considered untouchable, but bids for all other players will be entertained.
As a result, the likes of Bruno Fernandes, Maguire and Marcus Rashford could be let go if the right offer comes in, while Anthony Martial, Raphael Varane and Brandon Williams will all be leaving when their contracts expire.
Branthwaite's stellar 2023-24 campaign saw the 21-year-old register three goals in 41 matches for Sean Dyche's side, taking his total for the Toffees to four goals from 54 games across all tournaments.
The defender earned a spot in England's provisional Euro 2024 squad on account of his stellar campaign, but he was one of seven players axed when Gareth Southgate named his final 26.
What alternative centre-backs can Man United target?
If Man United's 'realistic' demand to Everton falls on deaf ears, one of two Sporting Lisbon defenders in Goncalo Inacio or Ousmane Diomande could be pursued instead, as well as fellow Primeira Liga man Antonio Silva of Benfica.
Crystal Palace's Marc Guehi and Lille's Leny Yoro are also of interest to the Red Devils, although there is a consensus that the latter will eventually end up at Real Madrid.
Furthermore, Man United have been barred from signing Nice's Jean-Clair Todibo by UEFA due to rules on multi-club ownership, as Sir Jim Ratcliffe's INEOS are also in control of the French club.