The 53-year-old, the overwhelming bookies' favourite to take over from Enrique at Camp Nou, is expected to be unveiled by Barca early next week.
Barcelona have been on the lookout for a new manager since Enrique, who still has a Copa del Rey final with Alaves next weekend to prepare for, announced in March that he intends to move on this summer.
Valverde has now taken a step closer to landing the job, taking him back to a club that he spent a couple of seasons with during his playing days.
A statement on Bilbao's official website read: "Ernesto Valverde will not continue as manager of the premier team of Athletic Club in 2017-18 season. The coach, accompanied by the President Josu Urrutia, will offer a press conference on Wednesday, May 24 at 12pm at Jose Iragorri press room of San Mames."
Valverde returned to Athletic for a second spell in 2013, guiding the club into Europe in his first three seasons but finishing down in seventh place in his final campaign.
Bilbao will still qualify for the Europa League should Barcelona beat Alaves in Saturday's cup final at the Vicente Calderon.