Premier League new boys Palace agreed an undisclosed fee for Gayle on Thursday and they were given permission to hold talks with the player.
Terms are thought to have been agreed and Gayle is now having a medical as he closes in on a move to Selhurst Park.
Gayle impressed at Peterborough last season after arriving initially on loan from Dagenham & Redbridge before making the move permanent in January.
The 22-year-old scored 13 goals in 29 games for Peterborough, but they were always facing a battle to keep him following their relegation to League One.
A number of clubs, including Fulham, have been linked with Gayle, but it appears Palace are on the verge of landing the former Arsenal youngster.
Palace are also close to signing former West Brom winger Jerome Thomas with the free agent also undergoing a medical at Selhurst Park as Ian Holloway looks to make the pair his first signings of the summer as he prepares for life in the Premier League.