The 29-year-old left the Nou Camp in the summer after four and a half years contracted to the Spanish giants.
He struggled to hold down a place there and made just nine starts for the Catalans, spending last season on loan to Greek side Olympiakos following a previous temporary stay at Schalke.
Afellay, who has 50 caps for the Netherlands, left Barcelona when his contract expired so the former PSV Eindhoven youth is a free agent.
Stoke tried to sign Yevhen Konoplyanka from Dnipro earlier this summer before turning their attention to Inter Milan’s Xherdan Shaqiri. They retain an interest in Dynamo Kiev winger Andriy Yarmolenko and Barcelona winger Adama Traore, but in remains to be seen whether the impending arrival of Affelay will put an end to those pursuits.