Depay, 27, is out of contract with French side Lyon at the end of June and has been a Barca target since Ronald Koeman was appointed last August.
Sources told ESPN on Tuesday that Barca are putting the finishing touches on a deal to make the former Manchester United forward the club's fourth signing of the summer.
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 finally backed by Laporta earlier this month, Barca have been able to press ahead with negotiations with the player's camp.
Meanwhile, Barca are hopeful of moving on several players as they look to make room in their squad and on the wage bill for their new signings.
Barca hope to receive around €5m for De la Fuente, while also retaining a percentage of any future sale, while Todibo will move to Nice for €8.5m plus add-ons.
Left-back Junior Firpo is also in talks about a transfer away from Camp Nou, with clubs in England and Italy interested, and Barca are open to offers for several other squad players.