The English Premier League club have reportedly paid the £10m release clause in the forward's West Brom contract.
According to Sky Sports News, the 29-year-old will return to Turf Moor ahead of the 2019-20 season following a superb campaign within which he netted 22 goals in 45 Championship appearances.
He was unable to help the Baggies back into England's top-flight as they lost in the play-offs, but will complete a return to the Premier League with a club where scored 41 goals in 128 matches.
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The former Southampton man will also be joined by Stoke City defender Erik Pieters at Burnley, reports suggest. The left-back spent last season on-loan at French Ligue 1 club Amiens.
Furthermore, West Brom are also set to lose Salomon Rondon on a permanent basis. The Venezuelan who spent last season on-loan at Newcastle United is set to link up with former Magpies boss, Rafa Benitez in China.