Aston Villa allowed Bjarnason - a semi-regular during last season's promotion-winning campaign - to terminate his contract ahead of the August transfer deadline, enabling him to sign for another club outside of the window.
As well as Derby and Stoke, the Iceland international is said to have interest from Nuremberg in the German Bundesliga.
But it is claimed that he is waiting for Stoke's managerial situation to reach a conclusion before he commits to his next move.
Joining either Stoke or Derby would ensure minimal upheaval after Bjarnason's time at nearby Aston Villa, although the chance to play top-flight football with Nuremberg may appeal.
Derby's need for an attacking midfielder could be enhanced by the suspension - or worse - that is likely to await Tom Lawrence after events this week.
Would the former Aston Villa man be a good addition for Derby or Stoke?
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