A transfer to Old Trafford would be a dream move for Rose and he was hoping that a £50m switch would be agreed after the World Cup.
Jose Mourinho has already brought 19-year-old right back Diogo Dalot in from Porto for £17.4m and it was expected that he would add another full back soon, with Juventus' Alex Sandro and Monaco defender Djibril Sidibe also linked.
But according to the Telegraph, the Portuguese manager has now advised Edward Woodward that other positions are a greater priority, with a new centre back top of the list.
United's interest in Rose could yet be reignited, with full back options remaining limited despite Dalot's arrival. Ashley Young was forced to play out of position last season, and Mourinho holds Luke Shaw in open contempt.
Everton have also shown an interest in Rose but a move to Goodison Park would be far less desirable even if the Toffees could muster the necessary funds to prise him away.
Rose expressed a desire to leave Spurs last summer, much to the chagrin of Mauricio Pochettino, but they reconciled after the 27-year-old decided to remain in north London.
Tottenham will be grateful if they can keep Rose for another year, as the man they want to replace him with will not be available until 2019.
Mauricio Pochettino had been eyeing a move for young Englishman Ryan Sessegnon but Fulham's promotion to the Premier League means that he is unlikely to be on the move this summer.