The 16 year-old spent time on trial with United earlier this season and has now completed a permanent switch to the club he supports.
Goss said: "The last few days haven't felt real - seeing all the first team players around and the training facilities is like a different world.
"As a Man United supporter as well as a (Exeter) City fan I couldn't believe it, I'd dreamt about it all my life.
"I can't wait to get started now and we've got pre-season games coming up now.
"I just have to continue working hard and my aim now is to just push higher and higher through the ranks.
"I've got family nearby and I'm sure it'll help a lot knowing they are just around the corner. City's will be the first result I look out for on a Saturday and I'll be doing my best to get to northern away games if I haven't got a fixture."