The 18 year-old joins the Gunners from Ajax's academy team.
Academy Manager, Per Mertesacker said: “We welcome Tommy to Arsenal. He is a young goalkeeper with big potential and we look forward to supporting his development in the coming years.
"The first step on Tommy’s development pathway with us will be to integrate into our under-21s academy group and train regularly with our first team goalkeeping group.”
Speaking after his deal was confirmed, Setford told the club's website: “I want to thank Arsenal for this opportunity. I can’t wait to hit the ground running.
“I already know Ethan (Nwnaeri) and Myles (Lewis-Skelly). They’re great guys and I can have a good laugh with them. I spent a few camps with them, so I know them well. I just saw them for a quick meet and greet!
“I’m looking forward to learning new things next season and pushing myself to another level. For the short term, it’s very important now to get a feel for the group and start to know everyone.