Manchester United are big admirers of the centre-back, who was recently selected in the England training squad for Euro 2024.
Recruiting in the heart of defence has been marked as a priority for the FA Cup winners - and the 21-year-old is high among the names being discussed regardless of who the manager is next season.
However, while also seeking a solid defensive midfielder and striker this summer, United need to box clever - financially - in the upcoming transfer window and have to bolster what they can spend with sales.
Branthwaite is thought to be valued in the region of £60m-£70m by Everton, who must sell one of their main assets before the end of June to stay in line with PSR.
Another player who could fetch a sizeable fee is Amadou Onana - but no move has yet emerged for the midfielder.
It means Branthwaite may become the more likely player to attract offers before June 30, and he has also been tracked by Manchester City and Tottenham.
Spurs looked very closely at signing Branthwaite before the January transfer window but ended up buying Radu Dragusin from Genoa, although they do want another centre-back this summer.
City may look at him if they lose a centre-back this summer but that is not expected to happen as things stand.
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