Nelson joined Arsenal as a youth player in 2007 and graduated from the academy last summer, making 16 appearances for the senior side across all competitions in 2017-18.
The 18-year-old has now put pen to paper on a new deal with the Gunners and will spend the first season of that contract on loan with Bundesliga outfit Hoffenheim.
"Reiss is a talented young player and we're pleased that he has signed a new contract with us," manager Unai Emery told the club's official website.
"This move to Hoffenheim will give him the chance to get regular high-level competition in the Bundesliga and is an important step in his development."
Nelson added: "I'm delighted. It's been a long time coming now, so I'm just happy everything's sorted out and I can sign a contract for a great club.
"It's my boyhood club. From a young age, from about eight years old, I've been supporting the club. The likes of Thierry Henry have encouraged me as well, so ever since then I've just loved the club and really fallen in love with it."
Nelson could make his debut for Hoffenheim during Saturday's Bundesliga clash with Freiburg.