United veteran Antonio Valencia is expected to leave Old Trafford this summer.
Ashley Young has played the majority of the season at right-back, with Diogo Dalot struggling to get regular game time.
And The Mirror say United are closely monitoring Calabria as a long-term replacement for Valencia.
Calabria, 22, has impressed in Serie A with Milan, making 26 appearances.
United boss Ole Gunnar Solskjaer hinted earlier this season it could be Valencia’s last year at Old Trafford.
He said: “He’s still not 100% fit, he is still working to get back fit and he’s had a fantastic career here at Man United and in England.
“He is one of the Premier League winners left in the dressing room but at the moment I’m not sure if Man United and Antonio will agree on the next year.
“So it depends if he gets back on the pitch as well in the next few months.
“But he’s the captain, a great servant to the club.”
United have impressed under Solskjaer since the Norwegian replaced Jose Mourinho at Old Trafford.
However, the Red Devils hero was handed his first Premier League defeat at Arsenal on Sunday afternoon.
Goals from Granit Xhaka and Pierre-Emerick Aubameyang handed Arsenal a 2-0 win.
The win moved the Gunners into the top four, ahead of United.
United are back in action on Saturday night against Wolves in the FA Cup quarter-finals.
The Red Devils then face Watford in the Premier League on March 30.