Manager Ole Gunnar Solskjaer is understood to have made the signing of a new midfielder one of his priorities for the upcoming window, in addition to a new winger and centre-forward.
Here, Sports Mole looks at how United could line up if the Spaniard arrived at Old Trafford.
Coupled with the January addition of Bruno Fernandes, the signing of Saul would leave United with one of the most star-studded midfield units in the Premier League.
The 25-year-old is capable of playing on either the left or right, but his main position is in the middle of the park, where he would slot in alongside Paul Pogba.
There has been speculation over Pogba's future, with Juventus and Real Madrid persistently linked with the World Cup winner, but Solskjaer has repeatedly stated that he expects the club-record transfer to still be at the club next season.
A midfield partnership of Pogba and Saul lacks an obvious holding player, with both eager to get forward when possible, but having played for so long under Diego Simeone there is no doubt that Saul would have the discipline to keep a defensive shape.
Solskjaer would have other options too; Fred and Nemanja Matic have turned their United careers around this season, while Scott McTominay was arguably their player of the year before suffering an injury in December.
Pogba and Saul would be the undisputed first choices, though, while in front of them Fernandes has already established himself as one of the first names on the teamsheet.
Jadon Sancho and Moussa Dembele have been named as targets at right-wing and centre-forward respectively, but for now the pace of Daniel James and Anthony Martial seems to be their best option in those areas.
Marcus Rashford has seemingly settled into his favourite position coming in from the left, and with a largely settled defence the only other position of major contention is left-back, where Luke Shaw and Brandon Williams are vying for minutes.
Questions have also been raised over the future of David de Gea given the fine form of Dean Henderson during his loan spell at Sheffield United, although the Spaniard still appears to be the undisputed number one ahead of next season.