The Gunners are understood to have agreed to sell the 26-year-old to the Turkish giants in a deal worth £7.5m plus add-ons, with the Uruguay international now deemed surplus to requirements by head coach Mikel Arteta.
Torreira was allegedly scheduled to board a flight to Istanbul on Tuesday ahead of undergo his medical and putting pen to paper on a three-year contract.
However, Daily Express journalist Ryan Taylor claims that the midfielder opted against travelling to Turkey and his proposed deal to join Galatasaray has now stalled.
Taylor adds that Torreira wants more time to make a final decision on his future, while both Arsenal and Galatasaray want answers as to why he decided against boarding his scheduled flight.
Earlier this summer, Torreira admitted that there is "no chance" of him staying at the Emirates Stadium beyond this summer.
Speaking about his future to El Telegrafo, Torreira said: "There is no chance (of staying at Arsenal). From the beginning, they told me that I didn't have a place, so I don't want to stay either. We'll look for a new direction."
The 5ft 5in midfielder has not made a competitive appearance for the Gunners since July 2020 when he came on as a substitute against Watford in their final Premier League game of the season.
Torreria has since spent the last two seasons away from the club on loan, firstly with Atletico Madrid for the 2020-21 campaign and then with Fiorentina last term.
The Uruguayan failed to make an impression under Diego Simeone as he started only three of his 19 La Liga appearances for Atletico on their way to lifting the title.
However, Torreira reignited his career with Fiorentina last season, starting 26 of his 31 Serie A appearances as well as contributing with five goals and two assists.
Despite a successful stint in Italy, La Viola decided against activating the €15m (£13m) option to buy Torreira on a permanent deal, before unsuccessfully attempting to drive his valuation down.
Galatasaray are still thought to be the frontrunners to sign the midfielder, while the likes of Valencia, Juventus and Roma have also been credited with an interest.
Torreira, who joined Arsenal from Sampdoria for around £27m in the summer of 2018, has less than 12 months remaining on his contract at the Emirates.