United boss Jose Mourinho is understood to have identified his midfield positions as an area which need improving, having seen injuries to the likes of Paul Pogba, Marouane Fellaini and Michael Carrick limit his options this season.
Mourinho has reportedly identified Saul as an ideal target to bolster his squad, with the 22-year-old having established himself as a first-team regular under Diego Simeone.
Saul only signed a new deal with Atletico last season, though, and United could be forced to pay the entirety of the Spain international's £134m release clause in order to prise him away from the Wanda Metropolitano.
However, the Red Devils are willing to use Ander Herrera as a makeweight in any deal in order to ease their financial outlay.
United have also been repeatedly linked with Saul's Atletico teammate Antoine Griezmann, although Barcelona are understood to be leading the race for his signature now.