The midfielder had been expected to clinch a transfer to the Serie A side after they reached an agreement with Pescara.
Italian newspaper La Gazzetta dello Sport reported at the weekend that the switch was on the verge of breaking down after Bjarnason's wage demands changed.
Leeds head coach Uwe Rosler worked with Bjarnason at Viking in Norway and Italian journalist Gianluca Di Marzio reported on his official website that Leeds were keen on the player at the start of June.
Tuttomercatoweb report Torino are becoming impatient with the player and have imposed the deadline, willing to move on if an agreement on personal terms cannot be reached.
The potential of the transfer falling apart gives a second chance to clubs interested in Bjarnason, and it could be that he has increased his demands because he has been offered a better deal elsewhere.
The central midfielder, 27, saw his Pescara side just miss out on Serie A through the play-offs, and agonisingly he could not play in the second leg of the final last month due to international commitments.