The 25-year-old has been strongly linked with a move away from Juve in recent weeks having struggled for regular football in the early stages of the 2019-20 campaign.
The Germany international has only made four Serie A appearances this season, while he was left out of the squad for the group stages of the Champions League.
According to Tuttosport, Can will be allowed to leave the Italian champions in the winter transfer window as he is not part of Maurizio Sarri's plans in Turin.
However, the report claims that United could face competition from the likes of Bayern Munich and Paris Saint-Germain at the start of 2020.
The Red Devils have lost Ander Herrera and Marouane Fellaini over the last two transfer windows to deplete their midfield options, while Paul Pogba has hardly featured this season due to an ankle problem.