Palmer has contributed to just five goals in 13 appearances this season and according to reports he is missing his friends back in Manchester which has sparked rumours about his future.
The England star has won Conference League and Club World Cup titles with the club and has established himself as a key player in recent years, but he could be on the move if the opportunity crops up.
Speaking to Card Player, 57-year-old World Cup winner Desailly suggested that Palmer may leave due to problems within the Chelsea side.
“If Cole Palmer decides to move away from Chelsea then that’s his business, but I just want him to respect the club, because they are the ones who have given him the tools to become the player he is now.
“I would love him to stay at Stamford Bridge because he’s got a key role in the team, even if (former manager) Enzo Maresca used him in various positions.
“I believe Palmer wants more stability and a system that’s built for him, but he still needs to fully recover from his groin injury and get back to full fitness. His recent performances speak for themselves, so he needs to stay humble and come back into the team to help the club, regardless of whether he stays or goes.”
Palmer has impressed sine joining Chelsea in 2023 from Manchester City but could be seeking a move back to the North of England where a host of clubs would welcome him with open arms.