Dest was reconsidering an offer from LaLiga side Villarreal after a phone call from coach Unai Emery, but sources say the 21-year-old is much more attracted by Milan's offer and is now on the verge of joining the Serie A champions.
Sources say the Italian side are in advanced talks with Barca over a deal worth around €20 million, which would see Dest move to Italy on a season-long loan with an option to make the move permanent next summer. Milan turned to Dest after losing Alessandro Florenzi to injury at the weekend and are able to offer Champions League football to the USMNT defender.
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.
Later on Wednesday, USMNT coach Gregg Berhalter said he was happy to hear about the possible move to Milan, as Dest figures to be an important member of his squad in Qatar.
"To go from Barcelona to AC Milan is not bad, right?" Berhalter said at an event to promote Nike's U.S. World Cup jersey launch on Sept. 15. "Serie A is a competitive league. You got to defend really well in the Serie A, and it's going to help him.
"It's not nice when you're told that you don't have a spot in the group. And Sergino is a guy that I know believes in himself and believes that he could keep fighting through it. But I guess this opportunity came up that he thought was good for him."
Manchester United, and a reunion with former Ajax coach Erik ten Hag, was Dest's original preferred destination, but the pieces did not fall 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. "I think [Antony and Martin Dubravka will be the final signings of the summer] and that will be it for this window, but when there is a great opportunity we always have to be alert at this top club."
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.