The 22-year-old only joined Peterborough in November last season and has caught Palace's attention with 13 goals, including an impressive hat-trick against Blackburn.
Gayle has made a rapid rise through the divisions, enjoying a brief prolific spell at Dagenham and Redbridge, and he now appears set to ply his trade in the Premier League with newly promoted Palace.
Peterborough director of football Barry Fry had confirmed earlier in the day that Palace were one of the clubs who were targeting the highly-rated youngster.
He told the South London Press: "We don't want him to go. He is on a four-year contract but so many clubs are in for him that it is very difficult for us to stop a player from League One moving to the Premier League.
"It has got to be a good offer - then we'd reluctantly let him go to the Premier League.
"He has got four years left but we are honourable people. We keep our best players as long as we can but if we get a good offer then we will listen to it.
"We've done that with all our players. The difference here is that Dwight is on a much longer contract.
"He is the best player I've had - by 100 miles. Just look at his goals - right foot, left foot and head."
Peterborough were condemned to relegation back to League One after a dramatic 3-2 defeat by the Eagles on the last day of the season.