During his loan spell with Championship side Middlesbrough last season, Doak made a positive impression, scoring three goals and providing seven assists in 24 games.
Liverpool included a buy-back option in the deal, similar to when they sold Jarell Quansah to Bayer Leverkusen.
The sale of Doak marks the latest of several big transfers for Liverpool this summer, with the club having raised a significant £227 million from player sales.