The Uruguayan midfielder could depart the Emirates before the transfer window closes next month as Arsenal look to bring in cash.
His stock remains high in Serie A, where he shone at Sampdoria, with Torino and Fiorentina making the running to secure his signature.
Both are eyeing loan deals but, according to Italian daily Tuttosport, Arsenal are only open to selling Torreira.
Arsenal want to earn a fee from his departure and close the book on his time at the Emirates Stadium, rather than sanctioning a temporary farewell.
Even a loan with an obligatory purchase option would mean Arsenal could need to wait at least 12 months to earn a fee from the Uruguayan’s exit.
Torino are still trying to convince Arsenal to accept a loan deal, with technical director Davide Vagnati working on the deal.
It remains to be seen if Arsenal soften their stance on Torreira and let him head back to Italy on a loan agreement.