Dest was reconsidering an offer from LaLiga side Villarreal after a phone call from coach Unai Emery, but is much more attracted by Milan’s offer.
Sources say the Italian side are in advanced talks with Barca over a deal which would see Dest move to Italy on a season-long loan with an option to make the move permanent next summer.
Villarreal had approached Dest about a season-long loan earlier this week after losing right-back Juan Foyth to injury, but the United States international was initially reluctant to accept their offer.
However, under intense pressure to leave from Barcelona, and given the need to play regularly before the World Cup, Dest decided to re-evaluate his options before the transfer window closes on Thursday.
Manchester United, and a reunion with former Ajax coach Erik ten Hag, would have been his preferred destination, but the pieces have not fallen into place for a move to Old Trafford.
“Aaron will stay in, so we will go [with this squad] from September to minimum January,” Ten Hag said in a news conference.
ESPN reported earlier this summer that Xavi Hernandez told Dest he would not feature regularly under him this season.
Despite having no other natural right-backs at his disposal, the Barca coach has said he prefers Ronald Araujo, Jules Kounde and Sergi Roberto there. Barca are also in the market for a new right-back.
Dest, 21, originally said that he wanted to stay and fight for his place, but after being left out of the squad for the first three games of the season he has been forced to rethink his stance.