The 16-year-old will join the Liverpool Academy until 2012 for an undisclosed fee.
The youngster will sign a professional contract on his 17th birthday.
“This is a great opportunity for Jamie,” said Swindon chairman Andrew Fitton.
“George Wood (Swindon’s goalkeeping coach) recognised his talent immediately and did a great job developing him.
“Being such a specialist position, the opportunity to go to an Academy like Liverpool is the right move for him and will give him the chance to play in tournaments all over the world.”
