It was reported over the weekend that United had made an initial offer of 12million euros for Darmian, who has also been linked with Arsenal, Bayern Munich and Napoli, with Everton apparently refusing to entertain selling Seamus Coleman this summer.
Cairo has now confirmed three separate clubs have bid for Darmian, although he would not reveal the interested parties.
“Three concrete offers have arrived for Darmian, but out of respect for those who made the offers I won’t reveal them, Cairo told Italian newspaper La Stampa.
“An extraordinary player like Matteo will want to try and win the Champions League and Toro can’t win it.”
Cairo has also warned clubs that any deal for the 25-year-old must be completed this week before Torino’s players return for pre-season training.
“Everyone who arrives at Bormio will stay for the rest of the year,” Cairo added.
“Extra time has begun and on Sunday there will be the final whistle, that is the deadline.