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But he understands competition for the 23-year-old is fierce, with Liverpool and Inter both believed to be interested.
Bowen said Stoke, 11th in the Premier League, would still try and sign the talented Switzerland international.
"To get a good player like him on board would be fantastic," he told BBC Radio Stoke on Tuesday.
"There are other clubs interested, but it doesn’t mean we rule ourselves out.
"He’s definitely on the list, but there is competition for a player of his ability."
Bowen said it would take some persuasion from Stoke's hierarchy to convince Shaqiri to join them, but believes Bojan's move to the club from Barcelona helps.
"It’s bound to help and it does help," he said.
"But there’s many boxes to tick, not least the financial side of it, and with Shaqiri coming from one of the best , if not the best, clubs in the world, there’s a lot of persuasion to get it over the line."
