Doncaster Rovers have now entered the chase for Imps keeper Burch, and have emerged as possible leaders of the pack.
The former Spurs keeper is under contract at City until the end of the season, after which he can depart on a free. Chris Sutton has tried to open contract talks, but is resigned to losing his prized asset for free in the summer.
He said: "Rob is going to want to progress his career as high as he possibly can.
"I would love to keep him because he is a fantastic player and he works hard.
"Talks are ongoing but I don't think you need to be a rocket scientist to work out what is probably going to happen."
With that in mind Sutton would rather pick up £20,000 in January than let him walk away free in May. Doncaster are chasing a keeper to take the place of their own veteran keeper Neil Sullivan, and may take the opportunity to jump the queue.
Of course Burch may spark a bidding war if he turns in a decent performance against Bolton in the FA Cup on January 2nd. That would leave Burnley trailing as they aren't prepared to pay cash for a keeper who won't get near the first team for several months.
Also believed to be interested at this stage are Wigan Athletic, although they are looking for a keeper to provide cover for Chris Kirkland rather than take his first team spot. Kirkland was injured at the weekend and ex Imp Mike Pollitt had to step in. However their interest at the moment is only mild.