The former Crewe midfielder, who has been in and out of the side since his arrival from Gresty Road in 2006, turned down a loan move to The Saddlers it has been reported.
The 'emergency' loan deadline window shuts on Thursday but Lunt would like to assess his options and try to push for a first-team spot at Hillsborough or move to another championship club.
It was also reported that Lunt could have joined big-spending Hyundai A-League side Sydney FC but that move broke down because of visa regulations and the new coach, John Kosmina, would rather add a striker rather than a midfielder to the squad.
Lunt's name, along with Lee Bullen and Peter Gilbert, have been made available to go on loan in a bid for Brian Laws to bring in new blood.
It is believed that Laws wants to move some players on to help fund a bid for Graham Kavanagh - who enjoyed a loan spell at Hillsborough earlier in the season from his parent club Sunderland.