A number of clubs have been chasing Bradshaw this summer with the player regarded as one of the best young goal-scorers in the country.
Bradshaw has scored 20 goals in each of his two seasons with League One side Walsall and the club have been facing a battle to keep hold of the player after missing out on promotion last season.
Barnsley and Rotherham United have both seen bids for the Wales international rejected this summer, while Rangers have also been linked with a move for the 23-year-old.
Bradshaw is understood to be frustrated by Walsall's stance to reject the offers from Barnsley and Rotherham with the striker keen to have the opportunity to play at a higher level.
Bradshaw is entering the final year of his contract at Walsall and he has turned down the offer of a new deal.
Barnsley have reluctantly accepted Bradshaw's request and it remains to be seen if his suitors come back with new bids for the striker.