The 23-year-old had a transfer request reluctantly accepted by The Red Imps in the spring and has now been targeted by Peterborough.
Futcher, whose contract expires next summer, is valued at £350,000 by Lincoln but Posh's bid falls considerably short of that price tag.
The League One side's manager Barry Fry confirmed his offer for the giant centre half and is refusing to concede defeat in his pursuit of the youngster.
"We have made a bid for Ben Futcher which is well below Lincoln City's asking price of £350,000," Fry told the Lincolnshire Echo.
"But we are hoping to go back with an improved offer and hope they reduce their asking price a little bit."
Lincoln chairman Rob Bradley declined to disclose details of Posh's bid but confirmed the club would not accept a derisory offer for Futcher.
Bradley said: "It is our policy not to discuss matters like this until there is something actually to say.
"We have adopted a degree of confidentiality and I would have hoped that would have also been done by any other parties involved.
"We accepted Ben's transfer request when he handed it in and he does hold a considerable amount of value to us.
"But any silly offers we get will be turned down instantly."
Meanwhile, Peterborough striker Andrew Fotiadis has had his London Road contract cancelled by mutual consent.
Fotiadis suffered a serious knee injury towards the end of last season and is unlikely to return to action before Christmas.
