The 39-year-old is understood to have been earmarked as the Reds' top target to replace Jurgen Klopp, who will leave his post after nine distinguished years at the end of the season.
Klopp announced his decision publicly in January two months after giving the club's hierarchy the heads-up, but Fenway Sports Group had a couple of loose ends to tie up before turning their attention to the German's successor.
Liverpool's owners managed to twist the arm of ex-Reds sporting director Michael Edwards, who is now working for FSG as their CEO of Football, while Richard Hughes has agreed to become the club's new sporting director.
With their upstairs appointments now taken care of, it is all systems go on the managerial search front, and Amorim left the door open to leaving Sporting when pressed on the Liverpool links in Friday's press conference.
Barcelona priced out of Amorim moveThe Reds have now been handed significant encouragement in their pursuit of Amorim, whom Sport claims is no longer on Barcelona's list of head coaches to replace Xavi this summer.
A matter of hours had passed from Klopp's exit announcement before Xavi surprisingly told a press conference that he would be leaving his Blaugrana post in the wake of a 5-3 loss to Villarreal in La Liga.
Barcelona chiefs have refused to throw in the towel in their efforts to keep Xavi, who has overseen a positive run of form since dropping his resignation bombshell, but the Spaniard remains adamant that he will be leaving the club.
Amorim naturally emerged as a contender for the Camp Nou gig, but the report claims that
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