The 16-year-old was in goal for the Young Lions when they made it to the final of the European U17 Championship in May.
Bursik was one of just two players in the squad not at a Premier League club at the time, but he will now join the top-flight Potters as a scholar.
Dons boss Neal Ardley is quoted by Sky Sports News as saying: "Sometimes these decisions are not just down to me and you have to look at what is best for the player and the club. Unfortunately, we have lost Joe, but he goes with our best wishes for the future.
"We must give [goalkeeping coach Ashley Bayes] credit for helping to bring Joe through. With Will Mannion also playing for England, we are proud to have had two young goalkeepers representing our country, and Bayzo has played a big part in developing both of them."
Bursik had been with the Dons since the age of eight and was part of their first-team squad against Swindon Town in April, when he was an unused substitute.