Despite some impressive displays for the Potters last season, Shaqiri was unable to help the club avoid the drop as Stoke slipped back down to the second tier of English football after a prolonged stay in the Premier League.
The report says that Liverpool have already opened discussions with Stoke over a deal for the former Bayern Munich and Inter Milan player, with Jurgen Klopp said to be keen to add him to his squad in time for the new season.
Shaqiri's World Cup campaign ended with Switzerland last week and he has indicated his next destination will be revealed soon.
Shaqiri was a major coup for Stoke when he signed from Inter in 2015, but despite moments of brilliance he has never shown his ability on a consistent basis in the Premier League.
Ojo, who spent previous year helping Fulham win promotion to the Premier League scoring four goals in 22 appearances despite being sidelined for a time by a dislocated shoulder, has impressed already in pre-season.
His side take on League Two outfit Bury on Saturday before a midweek game against Blackburn Rovers. Fekir, in any case, is still with France at the World Cup and would not be due to return to action until early-August.
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.