The publication claimed that the 45-year-old has already agreed personal terms with the Stadio Olimpico side and will say farewell to Milan this summer, but the Rossoneri have stressed there's no truth in the report.
"AC Milan categorically deny the information published by Corriere della Sera regarding its coach Massimiliano Allegri," a club statement reads.
Meanwhile, the newspaper also claimed that Inter has approached Allegri to replace Andrea Stramaccioni as the club's head coach this summer, but the Nerazzurri have issued a denial on their website, too.
"With reference to an article published in today's Corriere della Sera about the future coach of Inter, Inter categorically deny that there has been any contact - actual or requested - with AC Milan coach Massimiliano Allegri."
Allegri has been in charge of Milan since taking charge in June 2010 and, as of the start of May, his charges sit third in Serie A though Fiorentina is hot on their tails for the final Champions League berth.