Sportbladet says Stoke have won the tug of war for Johansson by triggering his buyout clause.
The goalkeeper is traveling from Arlanda to England this evening to complete his move to Stoke.
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Stoke finished 17th in the Championship this season.
Johansson kept six clean sheets in 46 competitive matches this season.
In total, there were 144 competitive matches in the Rotherham shirt for the Swedish goalkeeper.