More than 170 unwanted felines have been left with Cats Protection in recent months.
A recent kitten boom has left the city branch over run as fewer people are taking them in or adopting them.
It has caused the cat population across the UK to reach crisis point, according to animal welfare charities.
Cats Protection volunteers said they have been forced to turn away cats because they are full.
One woman said: "We have 174 cats in our care at the moment and a waiting list of another 150 cats.
"We're now having to turn people away. It's very sad - people just want to dispose of their pet instantly."
As well as an increase in kittens being born, a rise in strays and abandonments is being blamed on tough economic times.
Volunteers are now encouraging animal lovers to consider giving a rescue cat a second chance at happiness.
A recent UK-wide study, Tackling the Cat Crisis, claims the the number of cats entering the care system has increased from 29,200 in 2010 to 31,550 in 2012.
The report estimates the UK cat population is estimated to be between 9.5 million and 11.6 million.
Research found that many owners are delaying neutering, and animal welfare experts are calling on more owners to neuter their cats, as well as vets to promote neutering, to prevent unplanned litters.
Cats Protection is the world's largest feline welfare charity with more than 250 volunteer branches and 28 adoption centres across the UK. The Glasgow Branch is run by a team of unpaid volunteers.
If you want to adopt a cat call Glasgow Adoption Centre on 0141 779 3341 or visit www.cats.org.uk/glasgowac