The Colombia international returned to the Bernabeu last summer after two seasons on loan with Bayern Munich but has found it difficult to make his mark during the 2019-20 campaign.
Injuries have contributed to the fact that Rodriguez has only made 13 appearances in all competitions, although he has also struggled to earn a spot in Zinedine Zidane's side when fit and available.
Juventus, Everton, Arsenal and Wolverhampton Wanderers have also been linked with the South American, but it appears that Ole Gunnar Solskjaer's side are in a strong position to secure his services.
With that in mind, Sports Mole looks at the shirt numbers available for Rodriguez at Old Trafford should he arrive at Old Trafford during this summer's transfer window.
Alexis Sanchez currently holds the number seven shirt for United, but there is certainly more than a chance that the Chile international could leave on a permanent deal this summer. Sanchez is currently on loan at Inter Milan but has found it difficult to make his mark at San Siro, struggling with injury during his time at the club.
Should the experienced attacker depart permanently this summer then the number seven shirt would be available for Rodriguez, although there is also a chance that the famous jersey could be taken away from Sanchez even if he were to stay at Old Trafford or potentially leave on another loan deal.
The number 11 shirt is also available as Anthony Martial switched to the number nine ahead of the 2019-20 campaign following Romelu Lukaku's departure. Rodriguez actually wore the '11' during both of his seasons on loan at Bayern, but he currently has the number 16 for Real Madrid.
Number 16 at United is currently Marcos Rojo, who is back at Estudiantes on loan. The Argentine could leave on a permanent deal this summer, though, making that particular jersey another potential option.
Rodriguez has largely worn the number 10 shirt during his professional career, including for Colombia. Marcus Rashford currently holds the '10' for the Red Devis, though, and that is unlikely to change next season.
The Madrid attacker has also worn the number 19, number 8, number 13, number 33 and number five during his career at both club and international level and '19' is currently vacant in the United squad.
Third-choice goalkeeper Lee Grant wears the '13', making that another potential option, while there is also no '33' at Old Trafford, leaving Rodriguez with a number of options to choose from should he arrive this summer.