The Uruguay international was one of the first names on the teamsheet under Unai Emery but quickly fell out of favour upon Mikel Arteta's arrival at the Emirates in 2019.
Torreira also endured a dissatisfactory loan spell with Atletico Madrid before he was struck with personal tragedy after his mother died due to complications from COVID-19.
The 26-year-old's father pleaded with Arsenal to allow his son to return to South America, and he was widely expected to complete a transfer to Boca Juniors at one stage.
However, Torreira remained in Europe with Fiorentina and starred for La Viola during a season-long loan, banking five goals and two assists in 31 Serie A matches to help his side qualify for the Europa Conference League.
Fiorentina ultimately decided against triggering Torreira's £12.5m buy option, and the former Sampdoria man remains contracted to Arsenal for another 12 months.
Torreira has been expected to seal a permanent move elsewhere before the new season kicks off but could now be handed an Arsenal lifeline, as his agent Pablo Bentancur has revealed that the Uruguayan will be with the squad for their pre-season tour of the USA.
"Torreira would like to stay and play in Italy but as of today, Arsenal have called him up for the tour they will do in America, so it looks like he's part of their plans," Bentancur told SportItalia.
Arsenal have already beaten German second-tier side Nuremberg 5-3 in an exhibition match this summer and will face Everton, Orlando City and Chelsea during their tour of the States.
However, Torreira himself has insisted that he has "no chance" of staying at Arsenal, with the club informing him that he is not part of Arteta's plans for the upcoming season.
"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," Torreira told El Telegrafo. "We'll look for a new direction. Any team that wants me has to buy me, which would be around €15m (£12.9m)."
Torreira cost Arsenal £25.8m to sign from Fiorentina in the summer of 2018, and the midfielder has since registered four goals and six assists in 89 appearances for the Gunners in all competitions.
Torreira has won the FA Cup and Community Shield during his time with the Gunners, as well as becoming La Liga champion with Atletico in the 2020-21 season.
Lazio are thought to have been offered the chance to sign Torreira as Arsenal ramp up their efforts to agree a deal with Leicester City for Youri Tielemans.