The 35-year old goalkeeper is out of contract at the end of the season but has rejected two proposals from the Third Division champions in the midst of uncertainty surrounding what lies ahead for the Glasgow club.
"Negotiations have been going slowly," he told SunSport.
"The club has made it clear they want me to stay. They've offered me contracts but, with all the uncertainty over which division the club will be in next year, we're quite a bit away in terms of being able to agree a contract, unfortunately.
"I've been here for five-and-a-half years and I'm desperate to stay.
"My family are settled, my kids are loving life in school and we love life in Glasgow.
"We really want to stay but it looks like I possibly could be leaving."
When asked where he may continue his career, the former Cardiff City and Ipswich Town glvoeman said: "The Championship perhaps, the Premier League as a No2 or abroad. I want to keep my options open and don't want to close the door on anything."