Allegri agreed to return to Milan after more than 12 years last week after the removal of Sergio Conukicao.
Branchini said on Radio Rai: “We have analyzed the concrete and real proposal that we have received. The 'rumors' have hired excessive interest in our world. There is a hurry to anticipate events that can often lead to mistakes.
“Different clubs have approached Max Allegri in recent months, but we have to make decisions about what concrete materializes. The moment an interesting proposal comes to a club that likes the coach, an agreement is reached in a very short time. Often the imagination goes beyond reality.”
Asked if Roma had been in contact, he continued: “No, there was no offer. There was an attempt at contact many months ago, but we thought it was very unlikely as credible. There were no developments and there was not even contact.”
What is the role of Ibrahimovic?
On tension between Allegri and Milan director Zlatan Ibrahimovic, Branchini added: “I am not worried. I don't see the name of Ibrahimovic in the Milan Organizational Chart with a specific role.
“Ibra is a consultant for Redbird, but everyone in Milan has realized that there is a need for univocity, for understanding, for something that has been missing in recent months.
“I am not so worried, I hope the club made this choice, knowing that, in addition to changing a few players, some approaches to the players and the environment should change.”