The Uruguay international fell down the pecking order at the Emirates following the arrival of Mikel Arteta and also endured a difficult loan spell at Atletico Madrid in the 2020-21 season.
Torreira was also struck with personal tragedy last year following the death of his mother, and he was widely linked with a permanent return to South America with Boca Juniors last year.
However, the 26-year-old remained in Europe with Fiorentina and was an integral figure in the Viola midfield last term, scoring five goals and providing two assists in 31 appearances in Serie A to help his side make the Europa Conference League.
Torreira's loan deal at Fiorentina included an option to buy, but the Italian giants attempted to lower the €15m (£12.9m) clause, and a deal with Arsenal ultimately could not be struck.
The former Sampdoria man is therefore due to return to Arsenal for the pre-season training period, but he only has 12 months left to run on his contract with the North London club.
Torreira has now shot down any hopes of reviving his Arsenal career next term, telling El Telegrafo: "There is no chance. From the beginning, they told me that I didn't have a place, so I don't want to stay either."
The South American also personally revealed his price tag for the current window, adding: "We'll look for a new direction. Any team that wants me has to buy me, which would be around €15m (£12.9m)."
Torreira prepares to leave Arsenal having scored four goals and set up six more in 89 appearances for the club, winning the FA Cup and Community Shield during his four-year spell on the Emirates books.
After a proposed permanent move to Fiorentina fell through, Torreira expressed his dismay on social media and pointed the finger at unnamed individuals for failing to negotiate a transfer.
"Dear Fiorentina fans, unfortunately the time has come to say goodbye. It is with great sadness that my bond with Fiorentina is ended. A club that gave me so much love and support for all this time," Torreira wrote on Instagram.
"From the start, you made me feel at home. I discovered a marvellous city that was always kind to me, both on and off the field. Thanks to your trust, I was able to rediscover the consistency and form I wanted.
"I want you to know that I did my very best to continue being part of this club, but unfortunately some - acting in my view in a negative fashion - managed to prevent that from happening, so this is why I must leave."
Lazio are rumoured to be interested in signing Torreira this summer, with Arsenal already having Granit Xhaka, Thomas Partey, Mohamed Elneny, Albert Sambi Lokonga, Fabio Vieira, Martin Odegaard and Charlie Patino battling for starts in midfield.
Arsenal have been tipped to sign a new midfielder in the current window, but interest in Youri Tielemans cooled as the Gunners focused their efforts on signing Gabriel Jesus from Manchester City.
The 25-year-old has now completed his medical and should be announced as a new Arsenal player later this week, joining Vieira, Matt Turner and Marquinhos as Arteta's fourth summer signing.