The Netherlands international played a key role in his Dutch club winning a league and cup double during the 2018-19 campaign, in addition to impressing in their run to the semi-finals of the Champions League.
Van de Beek has again been in strong form during the current season, meanwhile, scoring 10 times and registering 11 assists in 37 appearances in all competitions.
Here, Sports Mole looks at the shirt numbers available for Van de Beek at Old Trafford should he arrive before the end of this summer's transfer window.
Van de Beek currently sports the number six for Ajax, which has been the case since he stepped into their first team. The '6' is occupied by Paul Pogba at Old Trafford, although the speculation surrounding the Frenchman's future continues to gather pace, and a summer exit for the 2018 World Cup winner cannot be ruled out.
Van de Beek wore '10' on a number of occasions for Ajax's youth and Under-21 sides, but it is unlikely that would be an option as Marcus Rashford is currently in charge of that particular jersey.
The midfielder's number for the Netherlands is '20' but that is also taken at United by back-up right-back Diogo Dalot. The Dutchman has also previously worn the number six and number 17 for his country, but the '17' at Old Trafford is held by Fred, who has been excellent in the second half of the season.
The number seven could be available depending on what happens with Alexis Sanchez this summer. The Chilean could stay and have '7' taken off him, though, which might open the door for Van de Beek.
Chris Smalling sported the 12 jersey for the Red Devils last season, but the centre-back could be sold this summer, potentially opening up another option for Van de Beek.
Elsewhere in the top 20, '19' is also vacant as it has not been filled since Rashford took the '10'.