The Reds have secured the signing of the young winger for a compensation fee of roughly £600,000.
The report adds that the 16-year-old had attracted interest from a number of clubs, including Leeds United, but the player's heart was set on a move to Anfield.
Celtic boss Ange Postecoglou handed him a senior debut against Dundee United in January as they tried to convince one of their brightest academy prospects to stay put. He also featured late on in February's Old Firm derby win over Rangers. However, Doak hasn't featured since after deciding to continue his career south of the border.
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