Jürgen Klopp is making his move to sign Shaqiri and wants to trigger his release clause, which is understood to be over £13 million, this month.
The Echo reports that Shaqiri would prefer a move to Liverpool and Stoke are hopeful that selling the Swiss global to the Reds would mean they would be more willing to loan out one of Harry Wilson and Sheyi Ojo.
Shaqiri's involvement at the World Cup ended last week when Switzerland were knocked out of the tournament by Sweden.
"Of course, I would like to play on the biggest stage in football".
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Klopp has already added two players - Naby Keita and Fabinho - to his squad this summer, but is aware of the need for further reinforcements if his side are to close the gap to champions Manchester City, and remain competitive in the Champions League too. He admitted that he would like to leave following last season's relegation.
Meanwhile, Shaqiri has made it clear that his future lies in the Premier League, having said: "You are going to hear soon on my future".
The winger, who scored seven goals for the newly-relegated Potters last season, has a release clause of £13.5m.
They could have him in place to travel to the U.S. where they will play Borussia Dortmund, Manchester City and Manchester United later this month. Fekir, in any case, is still with France at the World Cup and would not be due to return to action until early-August.