Atletico Madrid have also been linked with a move for the 26-year-old, who hit 13 goals in Serie A last season.
The former Fiorentina man helped his side to their highest finish in 22 years as they ended the campaign in seventh to secure a Europa League spot following Parma's exclusion.
Cerci has another two years remaining on his current deal with Torino, and Cairo is optimistic of holding onto the Italy international having already sold Ciro Immobile to Borussia Dortmund this summer.
Cairo said: "It's not just Milan who are interested, but other teams too. I would like to keep him. I still need to find out what the player's intentions are.
"We want to see whether there is a possibility of reaching an agreement with him or whether he is aiming for bigger clubs.
"We are playing in Europe next season, Milan are not."
Yes Torino are in European competition this season while Milan are not, but the player would have a better chance of winning trophies at Milan with a view to the next 5-6 years of his career that would seem to be the right move, I think he'd make a good addition to Milan myself- shea666