The England international has spent the past six seasons as a first-team regular for Spurs, scoring a total of 136 Premier League goals.
After failing to win any silverware during that goal-laden period, however, Kane recently admitted that he is not ruling out the prospect of a summer exit.
Signing Kane would mark a potentially pivotal moment in United's recent history, given that the club still lack a 25-plus goal-a-season striker required to challenge for the title.
As rumours regarding a move to Old Trafford gather pace, Sports Mole looks at the shirt numbers available for Kane should he join later this year.
Kane has played with the number 10 on his back at Tottenham since the start of the 2015-16 season following the departure of Emmanuel Adebayor.
The 26-year-old revealed at the time that he changed squad number in order to become a "club legend" at Tottenham, with 10 traditionally worn by prolific strikers of his type.
Unfortunately for Kane, though, that shirt number is already occupied by Marcus Rashford at Old Trafford and that is unlikely to change anytime soon.
Another number previously worn by the Englishman in North London, 18, was taken over by fan favourite Bruno Fernandes in January - again, that is unlikely to be up for grabs for the foreseeable future.
The number 37 shirt was worn by Kane for a short time, too, both in his early days at Tottenham and during a loan spell with Leicester City. Young midfielder James Garner is the player currently in possession of that one at Old Trafford.
Wearing the iconic number nine may appeal to Kane, with that his shirt of choice at international level with England, but that was handed to Anthony Martial ahead of this season after being relinquished by Inter Milan-bound Romelu Lukaku.
That leaves number seven, which had been occupied by Alexis Sanchez before he joined Inter on loan last summer, and 11 - previously Martial's before last year's switch.