ESPN reported on Sunday that Barca and Ajax had agreed on a fee for Dest, paving the way for the Catalan club to open talks with the right-back and his representatives over personal terms.
With an agreement over a five-year deal in place, Dest flew from Amsterdam to Barcelona on Tuesday, where he will undergo a medical before signing his contract.
European champions Bayern Munich had also been keen on signing Dest. Sources told ESPN they twice travelled to the Netherlands but were unable to strike a deal with Ajax.
Barca have admired Dest for a while but had to wait to sell Nelson Semedo before they were able to move for him. Once Semedo joined Wolves last week for an initial €30m, they stepped up their pursuit.
Ajax accepted a bid worth around €20m paid in a number installments from Barca, with another €5m due in performance-related add-ons.
Sources have told ESPN that the good relationship between the two clubs facilitated negotiations, with Barca also signing midfielder Frenkie de Jong from Ajax in 2019.
New Barca boss Ronald Koeman was also influential in getting the deal over the line. He has held a long-standing interest in Dest and when he was Netherlands manager he tried to persuade him to declare for the Oranje, but Dest opted for the USMNT.
Dest is expected to slot immediately into the first team set up Barca, although sources at the club have called for patience. The deal is unlikely to be finalised in time for Thursday's trip to Celta Vigo but he could be on the squad for Sunday's game against Sevilla.
The move to Camp Nou caps a remarkable rise for Dest, who has played just one full season of first-team football at Ajax, whose youth academy he joined in 2012.
Meawhile, U.S. youngster Konrad de la Fuente was named to the squad for the game at Celta.