The 30-year-old Scot, out of contract this summer, has been offered a new one-year deal but has missed the club's deadline of May 30 to respond and has yet to inform the Cumbrians of his plans.
"With Lee we're in a position where both parties have been away and linking up hasn't been possible at this moment in time," Abbott told the club's official website.
"It's one we do want to resolve and we'll seek to do that as quickly as we possibly can."
Abbott, meanwhile, is well on with plans to strengthen his squad and confirmed his interest in Gateshead winger Josh Gillies.
"Top of the list at the moment is a number nine, if Lee Miller doesn't sign, a creative right-sided player and someone who can perform the link role between midfield and attack," Abbott added.
"(Gillies) is fast, creative and has an eye for goal, and he's one of those players from that level who could be worth a calculated gamble.
"I think with Matt (Robson), Paul (Thirlwell), Danny (Livesey) and Sean (O'Hanlon) all agreeing new contracts it gives us a really good core of experienced players.
"They'll be able to help the young lads along and also help the new signings to settle in. It's key for us to have players like that when we're going to have a younger squad."