Alexander is out of contract at the Scottish Third Division outfit at the end of the season, and the former Cardiff City stopper has admitted that the uncertainty over which league the club will play in next season is hindering discussions.
The 35-year-old told BBC Wales: "Negotiations are going slowly.
"They have offered me contracts but with the uncertainty of what division the club is going to be in next season, we are quite a bit away, unfortunately.
"I love being at the club. That's five-and-a-half years and I'm desperate to stay. But it looks like I could be leaving. I'll keep my options open."
The Glasgow giants have won the league title, but the potential restructuring of the league pyramid in Scotland could dictate which division they play their football next season.