Wooster, 20, joins West Ham as a free agent after coming off contract at United over the summer.
After impressing in trials, Wooster has signed a one-year deal with a 12 month option.
“I had to bide my time, finding the right opportunity for me,” he explained. “I think this is the place to come and kick on. It's a big Club, and I'm so excited to be here.
“The lads and the coaches are top. I've already made a few good mates, and I love working with Robbo (Lead Coach Mark Robson), Pottsy (Assistant Coach Steve Potts), Billy (Head of Academy Goalkeeping Billy Lepine), and all the staff. I’m really enjoying it here so far.”
Lepine added, “You can tell Tom’s come in with the bit between his teeth to prove himself. His mentality towards every day has been fantastic to see. Clearly a very talented boy, comes from a good pedigree, has been at Barnsley, and has been at Manchester United before us.”