The Sun says Palmer is unsettled at Chelsea as he is missing family and friends in the Northwest.
Palmer, a Manchester United fan, came right through the Manchester City youth system before being sold to Chelsea three years ago.
And it's suggested the England international would be open to returning home and joining either Manchester club next season.
Aware of the speculation, Blues manager Liam Rosenior attempted to pour cold water on the rumours on Friday.
He said, “I’ve had numerous conversations with Cole and he seems very, very happy to be here.
“He is very, very happy to be here. He’s a huge part of our plans in the long term.
“He’s an outstanding player. Every player goes through difficult moments in their careers in terms of injury."
Palmer's deal with Chelsea runs to 2033.