The Athletic reports Doak, who has made first-team appearances against Dundee and in the Old Firm derby against Rangers this season, has found the prospect of working alongside Jurgen Klopp too good to turn down.
He is yet to sign a professional contract with Celtic so the Scottish side would only be due a minimal fee in compensation if a move to Anfield was completed.
"He's a fantastic kid who comes from a good family. He looked really composed for his age," Postecoglou said of Doak after his Celtic debut.
"He'll have grown up a bit today, it will given him a different perspective on life. He probably would have been on the other side of the terracing if he hadn't been playing!"
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