The Ligue 1 champions are understood to have contacted Rashford's representatives to sound out a potential move.
Rashford has entered the final year of his current deal at Old Trafford, although United retain an option to extend his terms for a further year.
PSG have long admired the United academy graduate and are keen to have a replacement for Messi already in place should the seven-time Ballon d’Or winner choose to leave next summer.
Messi’s existing terms are set to expire at the end of the season but include the option of a third year at the Parc des Princes.
In that instance, Rashford would be part of a reshaped attack built around Kylian Mbappe.
Rashford has endured a difficult year on-the-pitch, struggling for form and losing his place in the England squad as a result.
Despite once being considered a mainstay in Gareth Southgate's squad, the 24-year-old is currently unlikely to earn a place in the squad for the World Cup in Qatar.
Rashford scored just four Premier League goals for United in 25 appearances last season but has spoken of his desire of regaining form under new manager Erik ten Hag.