Developers have tabled an offer of more than £2m for the site, which the charity is considering, because the centre is struggling financially.
If the offer is accepted, the centre will be bulldozed, turned into a temporary car-park, and redeveloped into private or student accommodation – which would mean the loss of 10 jobs at the RSPCA.
Chris Whippey, the RSPCA's general manager for Leeds, Wakefield and District Branch, said that the money would be used directly to build a new animal centre on an out-of-town site.
A planned move to Rothwell Park fell through last year after asbestos was discovered.
"We are always on the look-out for green belt and farm sites but have not found one as of yet," he added.
Massive cuts have been made over the past 18 months to try and keep the centre open.
Eleven managers have been reduced to one, the veterinary service has been closed and dogs have been moved elsewhere.
But it appears that these measures have not been enough.
Mr Whippey said: "To stay at the site would need an investment of £70,000 to £80,000, into a building that is already 40 years old."
Animal licencing laws are against the site as well.
The minimum legal requirement is 400 yards between residential accomodation and a site containing dogs, but student accommodation is only 250 yards away.
This meant that the dogs, 88 at one point, had to be moved out to a sanctuary in Morley in June. Cats still remain at the centre, but they are being kept two to a cage, and nine to each exercise pen.
The centre has been allowed to carry on operating below licencing standards, under the condition that the RSPCA intended to move out at some point over the last few years.
Mr Whippey said: "There are a number of options available to us at the minute: dispose of the site, invest a substantial amount to bring it up to licencing standards or move the animals out and keep the buliding as a shell."
"There is a small window of time to decide on an option as the value of the site is at a premium at the moment and this must be taken into consideration," he said.
Mr Whippey spoke ofthe immense strain that
Leeds RSPCA, the third-oldest in the country, was under, inhaving to deal with the
highest rates of animal cruelty in any urban area in Britain.
This has led to a four-and-a-half-month waiting list forthem merely find a dog a place in care.
Other branches in Yorkshire have been in trouble inrecent times. Scunthorpe had to close due to vandalism, and the Sheffield branch is due to close as it stands in the way of a major road development.
Mr Whippey described the RSPCA as 185 branches each operating as an independent charity, responsible for generating their own income in their own areas.
"Many people perceive us as a cash-rich or reasonably wealthy charity, but this is simply not the case", he added.
Anyone wanting information, to adopt an animal or to make a donation should call Mr Whippey on 0113 2445182.