Sky sources understand the deal is worth £6m, but could rise to as much as £8m with incentives that have been included in the bid.
Bristol City manager Steve Cotterill told the club’s website after their Capital One Cup match on Tuesday night: “We’ve had a bid accepted and that’s as far as it’s gone.
“We need to speak to the player and his agent. I don’t think anything has changed whilst we’ve been playing, so maybe we’ll know more on Wednesday.
“He’s someone we’d like to have at the football club.”
Eagles boss Alan Pardew last week revealed that he was willing to let Gayle, who has reportedly been of interest to other Championship clubs such as Fulham and Wolves, leave Selhurst Park if the price was right.
He said: “There may be some players moved out but Dwight Gayle will not go anywhere unless his valuation is met.”
Gayle, 25, joined Palace from Peterborough in the summer of 2013 and has scored 18 goals in 24 starts.
For the Posh, Gayle had found the net 13 times in the Championship in 28 starts, having joined the club on an initial loan from Dagenham & Redbridge before signing for the club permanently in the 2013 January transfer window.
Gayle was a trainee at Dagenham and also spent time on loan at Bishop’s Stortford.