The 30-year-old shotstopper has been linked with a move away from Nuno Espirito Santo's team, although a suitor has yet to emerge with just one week left in the European transfer window. CBS sports has reported interest from Germany and Spain.
In 2023-24, Turner made 17 appearances across all competition for Forest, accumulating just two clean sheets.
During the summer international window, he represented the USMNT at Copa America, making three starts in the Group Stage of the tournament.