With the Gunners expected to wrap up a deal for Brentford's David Raya in the coming days, USA number one Turner will be free to leave in search of regular football.
It has been claimed that Arsenal and Forest have now reached a verbal agreement for the transfer of Turner, who was nevertheless named on the bench for Sunday's Community Shield win over Manchester City.
The 29-year-old is also believed to have accepted Forest's contract proposal, and talk of a move for Henderson has cooled as the Tricky Trees work to finalise the American's arrival.
However, NottinghamshireLive reports that Steve Cooper is still keen to strike a permanent deal for Henderson following his largely successful loan spell in the 2022-23 campaign.
The 26-year-old immediately became Cooper's first choice in between the sticks, keeping six clean sheets in 18 Premier League matches before he entered the treatment room with a thigh injury.
Paris Saint-Germain loanee Keylor Navas arrived in January to cover for the stricken Henderson, and the Costa Rican also impressed as Forest retained their Premier League status.
Navas has since returned to the French capital, though, and Cooper has subsequently been left with Ethan Horvath and Wayne Hennessey as his only two goalkeeping options.
As a result, the Forest manager supposedly wants to welcome two new shot-stoppers to the club this summer, although getting a deal for Turner over the line is their current priority.
Henderson would therefore face stiff competition for starts in a Forest jersey next season, but the England international is supposedly still keen to rejoin Cooper's ranks on a permanent basis.
Furthermore, Henderson has not featured at all for Man United in pre-season as he continues to recover from surgery, and he would play second fiddle to Andre Onana should he stay put.
The ex-Sheffield United loanee briefly ousted David de Gea as the Red Devils' number one in the 2020-21 season, but he quickly handed the gloves back over to the Spaniard and played just three games for the club in the 2021-22 season.
Henderson now has 13 clean sheets to show from 29 matches for Man United, whom he has represented since 2011, but he memorably blasted the club's "criminal" treatment of him last summer in the wake of his disappointing 2021-22 season, having been blocked from leaving on loan two years ago.
Forest are believed to be discussing a loan deal with an obligation to buy based on appearances for Henderson, who is out of contract at Old Trafford in the summer of 2025.
Cooper's side face a daunting trip to Arsenal in their Premier League opener on August 12, while Man United host Wolverhampton Wanderers to kick off the new season two days later.