Jose Mourinho has already made two signings this summer but is keen to acquire more players in the transfer window.
Tottenham are reluctant to part with the England global however, and have rebuffed any interest up to this point.
According to the Mirror, United are willing to offer Tottenham one final bid of £60m and will withdraw their interest in the England global if Mauricio Pochettino's side continue to resist the Red Devils' advances.
The Daily Mirror claim that United are willing to test the club's resolve one final time however, as they are considering lodging another bid.
Despite securing the services of defender Victor Lindelof and striker Romelu Lukaku, United have struggled in their pursuits of various midfielders, with Tottenham holding firm over Dier and Borussia Dortmund and Monaco holding off interest for Julian Weigl and Fabinho respectively.
Mourinho had hoped he could sign Dier for around £40m and an offer double his wages with the 23-year-old keen on the switch.