Officials from the Italian club were present to watch Evra in action during the 0-0 draw against Arsenal at the Emirates on Wednesday night.
Evra is out of contract at United in the summer, and is widely expected to depart Old Trafford.
Monaco and Paris St Germain have both been linked with offering the 32-year-old full back lucrative contracts to return to France.
However the Italians have now joined the race for his signature.
Evra has the same agent as Manchester United winger Nani, who Inter attempted to sign during the January transfer window.
Inter have already made a proposal to United captain Nemanja Vidic who has confirmed he will be leaving the club at the end of his contract in the summer, and the Serb is thought to be close to agreeing terms.
Inter remain keen on Nani, but would also be interested in Tottenham’s record signing Erik Lamela if Daniel Levy was willing to consider a sale or a possible loan move.
United are continuing to assess left back options of their own, with Southampton’s Luke Shaw and Atletico Madrid star Filipe Luis believed to be top targets.