Allegri agreed to return to Milan after over 12 years away last week after the removal of Sergio Conceicao.
Branchini said on Radio RAI: "We analysed the concrete and real proposal that we received. The 'rumours' have taken on excessive importance in our world. There is a rush to anticipate events that can often lead to error.
"Several clubs have approached Max Allegri in recent months, but we have to make decisions on what concretely materializes. The moment an interesting proposal materialises in a club that the coach likes, an agreement is reached in a very short time. Often, imagination goes beyond reality."
Asked if Roma had been in contact, he continued: "No, there wasn't an offer. There was an attempt at contact many months ago, but we considered it very unlikely to be credible. There have been no developments and there hasn't even been contact."
What is Ibrahimovic's role?
On tension between Allegri and Milan director Zlatan Ibrahimovic, Branchini added: "I'm not worried. I don't see Ibrahimovic's name in the Milan organisational chart with a specific role.
"Ibra is a consultant for Redbird, but everyone at Milan has realized that there is a need for univocity, for understanding, for something that has been missing in recent months.
"I'm not that worried, I hope that the club has made this choice knowing that, in addition to changing a few players, some approaches towards the players and the environment need to change."