The diminutive winger enjoyed a mixed World Cup; one that was peppered with moments of footballing brilliance and political controversy as Switzerland reached the last sixteen of the tournament, eventually losing out to Sweden.
There is a release clause in his contract - understood to be around £13m - and Liverpool are discussing the details on that with the club.
Former Stoke boss Mark Hughes signed Shaqiri from Inter Milan in August 2015 for £12million.
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Liverpool are also in talks with Stoke and are believed to be closing in on an agreement with the Potters over a fee for the winger.
Liverpool have already spent heavily in the transfer market this summer in the shape of midfielders Naby Keita and Fabinho and Jurgen Klopp is keen to strengthen his forward line with a mind to challenging for the Premier League title. Liverpool then fly to America for a three-game tour, where they will play Borussia Dortmund, Manchester City and Manchester United in the International Champions Cup. A deal for the Lyon captain may yet be resurrected after France's World Cup campaign, however.
Calls for Liverpool to sign a goalkeeper were intensified after Loris Karius' mistake against Tranmere this week, but the club have made no progress with regards to a move for Alisson Becker, their preferred target.