The Red Devils quickly settled on the 21-year-old as their priority defensive target for the summer window after his stellar season under Sean Dyche, where he contributed three goals from 35 Premier League matches.
Branthwaite's contributions at both ends of the pitch were key to keeping the Toffees afloat in the top flight, but their financial situation means that rivals are testing their mettle over some of their prized assets.
Man United apparently lodged an opening bid of £35m for Branthwaite, potentially rising to £43m with add-ons, but Everton swiftly rejected the approach and want £70m to consider a sale.
The Toffees' valuation has supposedly been met with a blunt response from Man United, who have told their Merseyside counterparts to "get realistic" if they want to strike an agreement.
Man United 'in no hurry' to make improved bid
Now, ESPN claims that the Red Devils will happily drop their interest in signing Branthwaite if Everton do not budge on their £70m price tag, as they are in no hurry to make an improved offer for the former PSV Eindhoven loanee.
Man United have apparently told Everton that they do not have the means to spend north of £70m on one player during the current transfer window, as they are believed to be working with an initial budget of just £50m.
The Red Devils' transfer kitty could be bolstered by player sales, but they still have no intention of bowing to Everton's demands unless Dyche's side relent and show a "willingness to negotiate".
From Everton's side, there are not under enormous pressure to let Branthwaite leave on the cheap, as he still has three years left to run on the contract extension he signed back in October.
Furthermore, the Toffees have reportedly received an enquiry from Arsenal over a deal for £50m-rated midfielder Amadou Onana, whose sale would lessen Everton's need to raise more funds through letting Branthwaite go.
Man United struggling to find affordable centre-backs
Man United dropping their interest in Branthwaite would see the Red Devils concede defeat in the pursuits of two of their defensive targets, as a move for Nice's Jean-Clair Todibo is a non-starter.
Due to UEFA rules on multi-club ownership, Man United cannot sign Todibo from their French counterparts - who are also under INEOS' control - a decision that Sir Jim Ratcliffe recently hit out at.
Alternatively, the Red Devils may look to pursue Crystal Palace's Marc Guehi, who has shone for England at Euro 2024 but would also cost a premium, potentially as much as £60m or more.
The same can be said for Benfica's Antonio Silva and Lille's Leny Yoro, although the former is said to be pining for a switch to Real Madrid either way, leaving Man United at risk of failing to sign any of their top defensive options.
Should Man United not have the budget to spare at all, a couple of notable names could be available on the free agent market, such as Mario Hermoso, Joel Matip and Mats Hummels.