Depay, 27, is playing for the Netherlands at Euro 2020, but speaking at a news conference on Wednesday, he confirmed his interest in joining Barcelona and hinted the move was close.
"Everyone knows I have been linked to Barca for a while now, and I want to play for Ronald Koeman," Depay said. "Just wait, and then the news will come."
Sources have told ESPN the move should be finalised by the end of the week, with both Depay and Barcelona having exchanged the necessary paperwork.
Depay would become their third free transfer this summer, following Eric Garcia and Sergio Aguero both signing for Barcelona from Manchester City. Barcelona have also paid €9 million to exercise a clause to recall right-back Emerson back from Real Betis.
Depay first agreed on personal terms with Barca last summer and a fee of around €25m was set with Lyon, but the club were unable to accommodate him within their La Liga imposed wage cap.
With no president in charge in January, Barca were unable to revisit a move for the Netherlands international, despite Koeman's desire to add him to their squad.
A move for Depay this summer was dependent on Koeman remaining in charge. After Koeman was backed by Joan Laporta earlier this month, Barca have been able to press ahead with negotiations with the player's team.