The Royals have now received clearance from the Football Association and US Soccer Federation.
London-born Stobbs played for Chelsea Ladies before enrolling at university in New York in 2012.
"Amber is a player we believe can compete in WSL 1 and score goals," boss Kelly Chambers told Reading's website.
Women's Super League One newcomers Reading have completed the signing of forward Amber Stobbs from US club Washington Spirit.