The Sunderland forward joined the Serie A giants on a loan deal last summer, with an obligation to buy this summer.
The terms of the deal meant the permanent move was effectively triggered within a few games of his AC Milan career and his agent has confirmed he will officially become a Milan player on July 1. The deal is understood to be around the £5million mark.
Borini's agent is Roberto de Fanti, Sunderland’s former director of football.
The Italian forward scored four times in 41 appearances last season for Milan, though he was often deployed in a wing-back role, and his form led to him being linked with a move to Juventus.
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De Fanti told the Italian media: "Yes, from the first of July he will be a player owned by the Rossoneri club until 2021.
"I would say Borini has had a positive season, given that there were low expectations when he arrived. He has proven to be worthy of Milan.
"Fabio is an exemplary professional and never seeks controversy, despite being played in several different roles across the pitch.
"He has worked very hard to be an important player for the club.
"He has never had any problems with Montella or Gattuso, adapting to their different tactical systems and philosophies. Clearly, Gattuso has switched from 3-5-2 to 4-3-3, which allowed him to return to an attacking position.
"I believe that the club are very happy with what Fabio has done this season. We feel the confidence from everyone. He has been a great signing. I think he will be a Milan player for a long time to come.
"Juventus? I think that such an adaptable player would be tempting for many clubs. After all, he has played for the likes of Chelsea, Liverpool and Roma – quality teams. The plan is to stay at Milan, as long as the club want him to remain."
Jeremain Lens' £6million summer move to Besiktas has already been confirmed after the Turkish giants exercised their option to sign the winger.
Lens has been on loan since last summer and the deal included an option for them to sign the Dutch winger permanently, an option that had to be exercised by the start of March.
Besiktas paid £6million, that fee includes the €1.5m loan fee that Besiktas paid last summer.
He will officially join Besiktas on June 1.