The 19-year-old centre-back, who plays for Switzerland Under 20s, will link up with Warren Joyce's Reserves after making his name in City's Academy side.
A versatile, cultured defender who is comfortable on the ball, Veseli arrived at Eastlands in 2008 from Lausanne Sport in his homeland, and has been tipped to eventually develop into a central midfielder.
"I am very happy and it is an honour to sign for Manchester United," the Swiss starlet said. "I can’t wait to get started, to work hard and do everything I can to help the team. I am looking forward to showing everybody what I can do on the pitch."
Veseli captained his country to victory in the Under 17s World Cup in 2009 and got his first taste of senior football during City's pre-season tour of North America last summer.
He'll now be looking to settle quickly at Old Trafford and impress for the Reds' second string in a bid to earn a first-team chance.
"I am very thankful to Sir Alex Ferguson for bringing me to such a great club," he added. "Manchester United has a long history of younger players coming through and hopefully I can follow and do the same."