Reports emerged last week suggesting that United had made a £50m approach for the England international, who has established himself as a key player for club and country over the past two seasons.
Spurs are unwilling to entertain bids for the 23-year-old, but The Mirror reports that United executive vice-chairman Ed Woodward is willing to go as high as £60m in an effort to prise Dier away from White Hart Lane.
United are making major changes to their squad this summer having only managed a sixth-place finish in the Premier League during Jose Mourinho's debut season, with Victor Lindelof and Romelu Lukaku having already made the move to Old Trafford.
A defensive midfielder is thought to be high on Mourinho's priority list, with Dier having been earmarked as an ideal target having faced competition for his place from Victor Wanyama and Mousa Dembele last season.
The player himself is understood to be open to the idea of a move, despite only signing a new five-year deal with the club in September.