The Red Devils were hoping to secure the Everton centre-half for a reasonable fee.
However, Manchester City boss Pep Guardiola may scupper those attempts this summer.
Per The Sun, City are also emerging as serious contenders for Branthwaite this summer.
Guardiola is impressed by how he plays out from the back and can handle the physicality of the Premier League.
Branthwaite was on loan at PSV Eindhoven two years ago, before coming back to start regularly for Everton last term.
