With Switzerland back from the World Cup after being eliminated by Sweden in the Round of 16, negotiations can progress and Shaqiri "has been considering his options since his involvement in Russian Federation was ended".
Shaqiri does have a release clause but the exact details are unknown at this stage, though the fee is believed to be higher than the widely reported £12m.
A move to Anfield would make Shaqiri the Reds' third signing of the summer following the arrival of midfielders Fabinho and Isaac Christie-Davies.
The Echo reports that Shaqiri would prefer a move to Liverpool and Stoke are hopeful that selling the Swiss worldwide to the Reds would mean they would be more willing to loan out one of Harry Wilson and Sheyi Ojo.
The Swiss worldwide will leave relegated Stoke City this summer, this much we knew already.
According to the Daily Mail, Liverpool's potential move for Shaqiri and Stoke's interest in Wilson are not linked in the slightest.
Klopp's side reached the Champions League final last season but finished a distant 25 points behind record breaking champions Manchester City. Fekir, in any case, is still with France at the World Cup and would not be due to return to action until early-August.