The 17-year-old is seen as one of the best young talents outside the Premier League and had already attracted interest from a host of clubs.
There were reports at the start of the month from the Daily Mail that Palace have offered Agyei a trial to try and beat Burnley to his signature.
However, Burnley have now made their move as they feel he will be ready for first team football next season.
Agyei is seen by the Clarets as the perfect long-term replacement for Danny Ings, who joined Liverpool earlier this summer.
The Ghanaian teenager has scored an impressive total of 35 goals for Wimbledon's Under-18 and Under-21 sides, and he is set to launch himself on the senior stage.