No financial details were given but British media said Forest will pay £7 million for Turner plus three million in performance-related add-ons.
Turner has earned 32 caps since his national team debut in February 2021 and also captained the US during their CONCACAF Gold Cup campaign this year. He won that tournament in 2021 and the 2023 CONCACAF Nations League.
Turner shot to fame in Major League Soccer where he made 111 appearances for New England Revolution before moving to Arsenal last year. Turner was backup goalkeeper to Aaron Ramsdale at the Gunners and made seven appearances.
"When I heard that there was interest to bring me here it just felt right for my family, it felt right when my agent spoke to me about it and it feels like the right move for me right now," Turner said in a statement.
"The manager (Steve Cooper) and the backroom staff, everything about the club has a little sparkle to it. It's felt right from the start and I'm happy it's got done."
Turner is Forest's third recruit of the close season after Ola Aina and Anthony Elanga. The American might even face his former side when Forest travel to Arsenal on Saturday for their first match of the Premier League season.
Arsenal are expected to replace Turner with Brentford keeper David Raya.
