The 22-year-old established himself as one of the Premier League's top young defenders last season, during which he made 35 league appearances and scored three goals.
United proposed two offers for the centre-back after the 2023-24 season concluded, with the first being in the region of £35m plus £10m in add-ons, while the second was £45m plus £5m in add-ons.
Both offers were turned down almost immediately, as Everton had set a valuation of £70m for the one-time capped England international.
Branthwaite arrived at Goodison Park from Carlisle United in the January 2020 transfer window and was sent on back-to-back loans to Blackburn Rovers and PSV Eindhoven before securing a regular starting spot under Sean Dyche last season.
Man United handed Branthwaite transfer boost
While formal negotiations over new terms between the Toffees and Brantwaite are yet to commence, The Daily Mail report that the 22-year-old is unwilling to accept less than the £160,000-a-week wage that United tabled earlier this summer.
The report adds that if the centre-back gets his wish, he will be in line for a significant increase, given that he currently pockets roughly £70,000-a-week.
It is understood that United remains interested in Branthwaite but he is not among their major priorities given his hefty price tag, and after the club secured the services of Leny Yoro.
Yoro, who is considered one of the most promising young centre-backs in Europe, finalised his transfer to Manchester last week after the club agreed on a £52m deal with Lille.
The Red Devils are also believed to still be in talks with Bayern Munich over a deal for Netherlands international Matthijs de Ligt.
Reports last week suggested that Erik ten Hag's side had tested the water with a bid of around £30m for the Dutch international, though it fell some way short of Bayern's valuation.
United still seeking defensive reinforcements
Bolstering their defence has been a major focus for United this summer, with Raphael Varane departing the club after his contract ended, and Willy Kambwala now sold to Villarreal.
Victor Lindelof is available for transfer, and the Red Devils might also consider selling Harry Maguire, who is now in the final 12 months of his current contract, if the right offer comes in.
Additionally, United are keen to acquire a full-back due to concerns about Luke Shaw's long-term fitness and Tyrell Malacia's ongoing recovery, as he has not yet returned to full training after a year-long absence.
Ten Hag's side have recently been linked with a move for former Chelsea star Marcos Alonso, as well as Crystal Palace full-back Tyrick Mitchell.