The Algerian has been on Arsene Wenger's radar for some time but a deal has never materialized and he remains at the King Power Stadium for now, but that could all change in the new year.
Mahrez was the talk of the last transfer deadline day as he pushed for a move away, with talk of interest from Manchester United and other clubs, but again a move did not happen and he stayed with the Foxes.
Reports in the Sunday press now claim that Wenger will offer him the chance of a move away in January, while there is also said to be interest from the club's arch rivals Tottenham.
Wenger has all but given up hope of keeping Alexis Sanchez and Mesut Ozil and is on the lookout for replacements, with Mahrez said to be one possibility for the Frenchman.
Leicester will fight to keep hold, but could decide to cash in if the player again pushes for a move and the offer on the table is big enough.