Coach mentions Gunners maestro, but Italian club's transfer chief is quick to water down the speculation as the need for a replacement for Andrea Pirlo grows in importance
The 24-year-old has long been linked with Barcelona, with the Catalan club's president, Sandro Rosell, keen to re-sign their former youth product.
Fabregas, though, has also been touted as a possible target for the Serie A champions, and Allegri has admitted that an addition in midfield is a priority.
"Having lost Andrea Pirlo, Milan need a player of important technical qualities," he told the press at an event in Livorno.
"We’ve not just discovered Fabregas this morning. There are two months of the market left and Adriano Galliani knows perfectly what this side is missing.
"If a player arrives then he will have an important value and will certainly be at Milan's level."
But Allegri's comments on Fabregas have been contradicted by a statement from club CEO Adriano Galliani, who has denied the Italian champions are following the Spaniard.
"Fabregas is not even a dream because we have never thought about him," he told Sky Sport Italia.
"There is zero possibility of seeing him at Milan."
Napoli's Marek Hamsik is another player who has been persistently linked with the vacant midfield berth at San Siro, but Galliani recently ruled out a move, saying: "Hamsik is a player of Napoli and we have never been interested."