Stubbington Ark has issued the threat in a bid to halt the tide of irresponsible people wanting to give up their pets.
However, the centre's manager Mike Ward assured The News that the threat was merely a deterrent.
Living quarters are over-flowing at the Ark
He said the centre would never put down a healthy animal, nor turn away someone in real need.
But with a record number of cats being dumped, and a dramatic fall in adoptions, he said accommodation at the centre was at bursting point.
Mr Ward said: 'We are really over a barrel. We have no choice, we get constantly blackmailed.
'Some people say: "If you don't take it, I'll leave it in a box by the road" and we can't say no.
'We say there might come a stage where we might be killing another animal to make room and they say it's not their problem. Their reaction is as cold and callous as that and a bit nasty and should not happen.'
The problem started in June when a record 145 cats arrived at the self-funding RSPCA shelter, in Ranvilles Lane, Stubbington, near Fareham.
Things are so bad the Ark is keeping cats in overnight drop-off units and an ambulance, while vets are holding recovering animals that they normally transfer to the centre.
The situation has been made worse as adoptions have fallen from the usual 10-15 a week to about three.
jeff.reines@thenews.co.uk