The USA international only joined Mikel Arteta's ranks last year from New England Revolution, replacing Fulham-bound Bernd Leno as the Gunners' number two.
With Aaron Ramsdale starting every Premier League match for Arsenal, Turner was restricted to minutes in the Europa League and FA Cup, playing seven games for the Gunners in total.
The 29-year-old kept four clean sheets in the 2022-23 season and earned plaudits for his solid displays, but he was never expected to challenge for the number one spot in North London.
According to The Athletic, Forest have now reached an agreement with Arsenal for the transfer of Turner, who will complete his medical at the City Ground later on Monday.
A separate update from journalist Ryan Taylor claims that Arsenal will receive around £10m including add-ons through the sale of the goalkeeper, who will join Ethan Horvath and Wayne Hennessey in the Tricky Trees' goalkeeping ranks.
Forest turned their attention to Turner after failing to make headway in talks with Manchester United for Dean Henderson, who spent the 2022-23 campaign on loan at the City Ground alongside Paris Saint-Germain's Keylor Navas.
Navas saw out Forest's end to the campaign as Henderson recovered from a thigh injury, but Steve Cooper's side are still thought to be pursuing the Englishman to compete alongside Turner.
Despite the advancing negotiations, Turner was part of the Arsenal squad for Sunday's Community Shield win over Manchester City, where Ramsdale played a crucial part in the Gunners' success.
The Englishman produced two fine stops from Phil Foden and Rodri before Leandro Trossard's late leveller, and also saved from the latter in the Gunners' 4-1 penalty-shootout success.
Ramsdale comfortably kept Turner at bay, but the American's exit will seemingly lead to Arsenal intensifying their efforts to sign Brentford shot-stopper David Raya.
The Gunners have apparently failed with an opening bid of £23m for the Spain international, which would have seen Brentford pocket £20m upfront and another £3m in add-ons.
Bees boss Thomas Frank has said that he believes Raya is worth £40m, but Arsenal are confident that they can drive the price down to around £30m, as the 27-year-old's contract expires in 2024.
If all goes to plan, Turner could make his debut for Forest and maiden Premier League appearance against Arsenal in their opening Premier League fixture on Saturday at the Emirates.