Former Leeds keeper Mervyn Day is also believed to be on the short-list, while Newcastle academy coach Peter Beardsley and Cheltenham's John Ward have been linked with the position.
But the Cumbrians see Carver as the ideal replacement for Paul Simpson, who left to take charge of Preston last week.
Carver was expected to be promoted to assistant boss by Leeds this summer after Sam Ellis parted company with the club.
If he does leave, Leeds would be short of coaching staff with the start of pre-season training only nine days away, but Carver, who joined Leeds' backroom team 12 months ago, admitted today that he was undecided over his future.
Carver said in the Yorkshire Evening Post: "I've had a great year with Leeds and that would make leaving the club a very difficult decision.
"I'm considering my options and I haven't decided on anything. I want a bit more time to have a think about the situation."