Fellow defenders Bobby Hassell and Darren Moore are also struggling with injuries, leaving Davey short of options at the back.
Davey told the club's official website: "Luke has suffered a torn meniscus in his knee which is a big blow for him and everybody else. I thought he had an excellent game on Tuesday night but afterwards his knee swelled up badly.
"He will go to see a specialist this week and will be out for up to six weeks after his operation. Bobby Hassell is struggling for the weekend and we do have a doubt again over Darren Moore. We will have a look at things in the next 24 hours and could be forced to go into the loan market."
On a more positive note, new signing O'Neil Thompson has been granted a work permit and will be available to make his debut once he has received international clearance.