The 53-year-old was apparently a favourite to take over from Gerardo Martino in 2014, but the club opted for Enrique instead.
The Barcelona president, Josep Maria Bartomeu, said: "Valverde has the ability, the judgment, the knowledge and the experience".
Valverde began his coaching career working with youngsters at Bilbao Athletic, Athletic Bilbao's youth team.
Barcelona have yet to replace Dani Alves, who joined Juventus last summer, while Valverde could also look at signing a new centre-back amid uncertainty over Javier Mascherano's future.
Enrique had announced in March he would be leaving the club at the end of the season to take a one-year break from management.
Barcelona's golden generation saw players such as Messi, Xavi Hernandez, Andres Iniesta, Sergio Busquets, Carles Puyol and Gerard Pique come through the La Masia youth system. "He is a hard worker and is passionate about using technology in training and in match management".
What do you make of the appointment of Valverde?
Herrera was voted the club's player of the season and he won his first cap for Spain against England at Wembley last November.
Valverde led Athletic to victory over Barcelona in the final of the 2015 Spanish Super Cup, which marked the team's first title in more than three decades.
Jose Mourinho has been hugely impressed with Herrera's attitude and work-rate since his own arrival at the club last summer and is ready to offer the Spaniard a bumper new deal to keep him at Old Trafford. "He should enjoy it", Luis Enrique said when asked what advice he would give Valverde after a Messi-inspired 3-1 win over Alaves in his final game in charge.