The former Charlton midfielder is the target for the Scottish champions and current league leaders - but they only want him on loan - something that might appeal to player and Chelsea alike.
The Evening Times, in Glasgow, splashed with the story and hinted that Parker could be moving north quickly.
With little money to spend, Celtic manager Martin O'Neil is looking for loan deals, and Scottish rules allow him to sign a couple more players for the second half of the season - and Parker fits their bill.