The Daily Mail says the former Reds sporting director accepted the offer at second attempt after talks in Boston with FSG boss John W Henry.
Richard Hughes will come in as sporting director after leaving Bournemouth last week.
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Edwards and Hughes know eachother after their time at Portsmouth, where the former was a data analyst and the latter a player.
Edwards left Liverpool in 2022 after ten years with the club and has stayed out of the game since then.