The club believe they can convince Edwards to take on a slightly different role to the one that he left previously.
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With manager Jurgen Klopp leaving and sport director Jorg Schmadtke leaving, Liverpool do have holes to fill.
Edwards is likely to come in as a technical director or some other position, as he does not want to work as a sporting director.
Mike Gordon, president of Liverpool's owners Fenway Sports Group, has been talking to Edwards about a return.