An unnamed League One club have put an offer in writing but it has been rejected by the Cumbrians.
Carlisle’s managing director John Nixon told the club’s official website: “We have received an official written offer from a League One club for Kyle and we have kept the player informed along the way.
“We spoke to the manager and the player about the approach and I can confirm that we have turned the offer down.”
Dempsey, 19, scored 11 goals in 44 appearances after breaking into the first team last season and has been monitored by several clubs.
He was linked with a move to Burnley last season and Carlisle also turned down a bid from Barnsley at the turn of the year.