United have been heavily linked with a move for a new left-back this summer, having used Ashley Young as a makeshift defender in that position for the majority of the 2017-18 campaign.
However, manager Jose Mourinho is understood to have been suitably impressed by Young's performances - and the presence of Luke Shaw as a backup - to have made finding a replacement less of a priority.
The Red Devils have already reportedly shelved plans to sign Alex Sandro from Juventus, and The Telegraph reports that Rose has also now been relegated in their list of summer targets as Mourinho instead looks for a new central defender.
Rose was understood to be keen on a proposed £50m move to Old Trafford, but could now remain at Tottenham despite vying for first-team football with Ben Davies last season.
The England international has also been linked with the likes of Everton, Juventus and Borussia Dortmund in the past.