With Luke Shaw still recovering from the foot injury that ended his campaign prematurely last season, Jose Mourinho has deployed Matteo Darmian, Daley Blind and Ashley Young at left-back so far this term.
According to the Sun, the Portuguese coach is determined to bolster his options in the position and has identified Rose as the ideal candidate for the role, largely because of his attacking abilities.
However, the England international could prove a difficult target to secure, as Spurs will likely be reluctant to sell one of their best players to a direct rival and Mauricio Pochettino is unlikely to be willing to weaken his squad halfway through the campaign.
Additionally, United would have to convince chairman Daniel Levy to do business with them. Levy reportedly vowed never to negotiate with United again, after the 20-time champions of England hijacked Dimitar Berbatov's proposed move from Spurs to Manchester City on deadline day in 2008 and secured the Bulgarian.
Levy stood by his principles in the summer, when United were forced to look elsewhere after their courtship of Eric Dier proved fruitless.
In his 10-year spell in north London, Rose has played 144 games for Spurs and has won 12 England caps.
In August, Rose was forced to apologise after demanding the club sign "well known players" who you don't "have to Google" and insisting he would "make sure I get what I am worth" with regard to his salary which stands at £65,000-a-week. In a wide-ranging interview, Rose also revealed his intent to play up north to be closer to his family and that "time was running out" if he wanted to win silverware.
"Time is running out and I do want to win trophies," he said. "I don't want to play football for 15 years and not have one trophy or one medal. Sorry, that's not what I am about. I wouldn't be happy with that. I want to win something. I will say this too — I will play up north. I don't know exactly when, but I will get back up north and play some football somewhere. I moved away from home (from Leeds to Spurs) at 16."
Curiously, Rose was left out of Spurs' official 2018 calendar, as he is not appearing on the cover nor on any individual pages. However, it appears there is a valid reason for the England international's omission.
"Danny is not the only player who doesn't appear in the calendar – Ben Davies, Erik Lamela and Moussa Sissoko are not in it either, along with our summer signings," the source was quoted as saying by the Sun.
"People will say it doesn't look good for Danny but the fact he has not played this season means there are no images of him in our new Nike kit that could be used in the calendar. I am sure, that after he returns to action, he will be in the 2019 calendar – assuming he is still with us, obviously."
Chelsea and City are also reportedly keeping tabs on Rose, who only returned to training on Wednesday (11 October) having not played since 31 January where he suffered a season-ending knee injury away to Sunderland.
Along with Lamela, Spurs expect Rose to be fit and available for selection for the north London derby against Arsenal on 18 November.