Dest, 19, will fly to Barcelona for his medical after agreeing personal terms as Ronald Koeman closes in on his first signing as Camp Nou boss.
Koeman has had a long-standing interest in Dest. When he was Netherlands manager, he tried to persuade Dest to declare for the Oranje, but Dest opted for the USMNT.
It's been a remarkable rise for Dest, who has only played one full season of first-team football at Ajax.
But as Koeman starts his Barcelona rebuild, he has turned to Dest as the first piece of the jigsaw. Barca's interest in Dest was always dependent on players leaving Camp Nou.
Nelson Semedo's departure to Wolverhampton Wanderers freed up transfer funds to allow Barca to make an official approach for Dest and, after four days of negotiations, the two clubs have now come to an agreement.
Information from Barcelona correspondents Moises Llorens and Sam Marsden was used in this report