Sir Alex Ferguson has moved to snap up the 16-year-old after he impressed on trial for the Red Devils during a tournament in the Netherlands earlier this year.
Goss joined Exeter's centre of excellence as an eight-year-old and scored 13 goals from midfield for the club's Under-16 team last season.
The League Two prospect professed to growing up as a United supporter and revealed he was overwhelmed after sealing a deal to join his boyhood club.
"The last few days haven't felt real - seeing all the first team players around and the training facilities is like a different world," Goss told Exeter's official website.
"As a Man United supporter as well as an [Exeter] City fan I couldn't believe it, I'd dreamt about it all my life.
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"I can't wait to get started now and we've got pre-season games coming up now."
Goss is now focused on progressing at Old Trafford but insisted that he would continue to keep an eye on the fortunes of his first club.
"I just have to continue working hard and my aim now is to just push higher and higher through the ranks," he continued.
"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."