The 22-year-old played 15 times for Bristol City last season on loan, but now he is set to rejoin the Championship side on a permanent deal.
Chelsea and Kasey Palmer wanted stability and gave Palmer the go-ahead to leave the club on a permanent basis.
As per Football Insider, Palmer arrived at Bristol City for a medical on Wednesday afternoon as Swansea City appear to have missed out.
The Welsh side were wanting a loan deal, but with Chelsea and Palmer's stance, a deal became increasingly unlikely when it looked on as Steve Cooper's influence was significant in persuading Palmer to head to the Liberty.
But Lee Johnson's side made the first move, and the desired move, and Palmer looks set to be the third Blue to join the Robins this summer.
Jay Dasilva and Tomas Kalas have already secured moves to the south-west earlier on this month.