The 26-year-old Uruguayan is reportedly being sold by the Gunners to the Turkish giants for a reported £7.
5 million plus add-ons after head coach Mikel Arteta decided the player was no longer necessary.
According to reports, Torreira was due to travel to Istanbul on Tuesday to complete his medical and sign a three-year contract.
Ryan Taylor, a journalist for the Daily Express, asserts that because the midfielder decided against visiting Turkey, his proposed transfer to Galatasaray has since come to a standstill.
Taylor continues, "Torreira wants more time to make a decision about his future, while Arsenal and Galatasaray want to know why he decided not to board his flight."
Torreira acknowledged earlier this summer that there was "no chance" he would continue to work at the Emirates Stadium after this season.
There is no chance that I will stay at Arsenal, Torreira said when asked about his future to El Telegrafo. Since they informed me right away that I didn't have a place to stay, I decided against doing so as well. 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 €15 million (£13 million) 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.