Cresswell was brought to West Ham by the club’s former boss who has a problem at the position.
Allardyce signed Liverpool born Cresswell who has gone on to make over 150 appearances for the Hammers since 2014.
The Daily Mail claims the 28-year-old could be tempted to return to his home town despite him being a lifelong Liverpool supporter.
The Hammers may be ready to listen to offers in the summer with many believing the player has not been able to catch the form of recent seasons although he has performed decently in the new 3/5 at the back system set up by David Moyes since his arrival.