The 29-year-old's contract at Brunton Park is due to expire during the summer and despite having held talks over fresh terms, no extension has been signed.
"I like Lee Miller, and so do the players. I think the fans would like to know if he will be with us or not, but he'll make his decision at the right time for him," Abbott told the club's official website. "We have to hope that he'll give us another 12 months because I think he's an important player for us.
"I talk to him regularly about his situation but he has to make a decision that works for him. He has told me that he's serious about staying but that he wants to see what's around and he wants to think about what will be the best thing for him to do next season.
"You have to allow players to do that. That's the way the game is and we have to wait to see what comes of it."
Miller joined Carlisle from Middlesbrough back in 2011.