According to vice-chairman Bob Dorrian the new deal for Burch would make him the highest paid player the the club 'for years to come'.
However its likely that Burch will leave anyway with strong interest from Doncaster, Bolton and Wigan. It's also rumoured that Everton may now have entered the race for the stopper.
"We have made him an excellent contract offer,"said Dorrian. "He has not officially rejected it, but we don't expect him to accept it."
If Burch is to move in January it could mean a small payday for The Imps as a side buys out the remainder of his contract. However should he hang on until May he'll walk away a free agent with a lucrative signing on bonus as well.
"We haven't started discussing a valuation for him," added Dorrian "Whether we accept an offer depends on a couple of things – the size of the offer and our league position.
"Although we don't want to lose him for nothing, Rob could be a key player in helping us stay up. It's very much a balancing act."
There is another scenario which a Premiership side may choose to exploit. If they were to pay a nominal fee in January (lets say £50,000) then they may be willing to loan Burch back to City for the final five months of the season, and take him full time at the end of the season. Although there is a remote chance of this, it may be a route that a bigger side go down in order to secure their man.