Edwards, 43, has been the architect of the Reds' success in the transfer market that has seen them become the envy of their rivals.
In November, he announced that he would be leaving his role as sporting director at the end of the season to spend more time with his family.
But both United and Chelsea are considering him as they go through important transition periods, according to The Mail.
The Red Devils are in the middle of overhauling their recruitment set-up, with Richard Arnold replacing Ed Woodward as chief executive.
John Murtough was hired as football director and Erik ten Hag has been installed as the new manager tasked with changing their fortunes.
Chelsea are also keen to make Edwards their new sporting director following the takeover by Todd Boehly's consortium.
The Blues confirmed on Wednesday that Marina Granovskaia would leave after spearheading the club's transfer business in the Roman Abramovich era.
Edwards joined Liverpool in 2011 as head of analytics before working his way up to sporting director.
He became a pivotal figure at the Anfield club and brought in the likes of Alisson, Virgil Van Dijk, Mohamed Salah, Sadio Mane, Andy Robertson, Thiago and Fabinho.
He also raised phenomenal sums from player sales, including £146million when Liverpool sold Philippe Coutinho to Barcelona.