Ojo, who spent past 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.
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.
Shaqiri would not be seen as someone to come in and start games immediately, though Klopp appreciates the importance of depth across his attacking positions.
Stoke are resigned to losing the Switzerland worldwide, who understandably wants to stay in the Premier League after last season's relegation into the Championship.
A good World Cup for Switzerland means Shaqiri still has top clubs eyeing him in the window despite his less than stellar record at Stoke over the last couple of campaigns.
Liverpool may be able to sweeten any deal by offering a player on loan.
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.