Allegri closer to becoming Milan coach
"If he frees himself from Cagliari it will definitely be Allegri," Milan chief executive Adriano Galliani told reporters on Thursday.
The 42-year-old Allegri, viewed by many commentators as Italy's best young manager, emerged as front-runner for the job last month.
He was surprisingly sacked by Cagliari after a row in April but remains contracted to the Sardinian club.
"I'm not losing patience," Galliani added. "Allegri is a coach linked to Cagliari until June 30, 2011 therefore it doesn't depend on us but Cagliari president Massimo Cellino.
"He is now in Miami. I will see him next week but our personal relationship is good."
If the deal with Cagliari is resolved, Allegri will take over from Leonardo who left Milan by mutual consent at the end of the season following a third-place finish in Serie A.