Shaqiri was among Switzerland's best players in their run to the last 16 of the World Cup, scoring a late victor in their only victory, against Serbia.
A product of the Basel youth system, Shaqiri has spent the last three seasons at the Brittania, coming over from Inter for a then-club record £12 million.
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The 27-year-old has two years remaining on his current contract at Stamford Bridge and has not yet committed to a new deal. Nevertheless, losing a player of his talent would be a devastating blow to the Blues and arriving manager, Maurizio Sarri.
Jürgen Klopp, the Liverpool manager, believes that Shaqiri, who is on holiday after playing at the World Cup, would be a valuable addition to his squad, having followed the player when the Swiss was with Bayern Munich.
Shaqiri has indicated that he wants the move which should be sealed before the Reds head off to the United States as part of their pre-season plans. Liverpool saw a deal for Lyon captain Nabil Fekir break down due to a medical issue last month, but serious alternatives are yet to emerge amid claims from the player's agent that the move remains on the table.