The Serie A champions have an option to make the deal permanent next summer for €20 million.
ESPN reported earlier this summer that Barca coach Xavi Hernandez had told Dest he was no longer part of his plans.
Dest, 21, initially said he wanted to stay at Camp Nou but changed his mind after being left out of the squad for the opening three games of the season.
The defender is set to feature for U.S. at the World Cup, which begins in November, and he decided the offer from Milan, who will play in the Champions League this season, was too good to turn down.
Villarreal had also made a move to sign Dest on loan, with coach Unai Emery even phoning him Tuesday, but Dest opted to move to Italy instead.
Milan’s interest only arrived Wednesday after they lost right-back Alessandro Florenzi to injury.
Dest’s camp was contacted by the Italian side in the late afternoon and by the evening an agreement was in place, with the player flying to Italy on Thursday to complete the deal.
Barca signed Dest for €20m from Ajax in 2020 and he had an impressive debut season under former head coach Ronald Koeman, making 41 appearances in all competitions.
However, injuries and a change of coaches — Xavi replaced Koeman last November — led to Dest losing his place in the side in the second half of last season.
Despite that, Xavi had been willing to give him another chance this season at the start of the summer, but the club’s need to reduce the wage bill and raise money to balance their spending forced them to change their stance on Dest.
Meanwhile, Barca forward Memphis Depay confirmed that he would not be leaving Camp Nou before the end of the transfer window.