If you are looking for a yes–no answer, then it depends on what you mean by “correct”. Neither of your two sentences is particularly natural, but neither is ungrammatical, either.
In other words, they are “grammatical” but they may not be correct for your situation. In fact, I suspect that neither is a good match for the general case. That doesn’t make them ungrammatical, but being grammatical and being fitting or reasonable or desirable or correct, let alone being optimal, are all different things entirely.
To expand and explain, the following closely related sentences below are all grammatical — or at least can be under the right circumstances. They are not all equivalent to one another, however. Most differ in nuance, or perhaps in register.
There isn’t a cat in the kitchen.
This is probably the most marginal of the set, but could possibly be made to work with sufficient set-up. Compare:
- There isn’t a clean dish in the whole damn house.
See number 4.
There isn’t any cat in the kitchen.
Again, this sounds a bit odd, but this time it is because it seems to be construing cat to be a mass noun instead of a count noun. Compare:
- There isn’t any milk in the fridge.
There’s no cat in the kitchen.
Unlike the earlier pair, this one is perfectly fine. It seems to answer the question,
- Is there a cat in the kitchen?
There’s not a cat in the kitchen.
This is equivalent to #1, and so would not sound right under most but not all possible circumstances. It suggests questions like
- Well, what is in the kitchen then?
- You have a cat in every room of your house, don’t you now?
There are no cats in the kitchen.
This one is perfectly fine, and is fully equivalent to the next one, but for the slight possible register shift triggered by the contraction.
There aren’t any cats in the kitchen.
Again, sounds just fine. It appears to answer the question
- Are there any cats in the kitchen?
No cat’s in the kitchen.
Here again is a rarer one. But perhaps you’ve got a cat in the den, and in the foyer, and in the pantry, but no cat’s in the kitchen.
No cats are in the kitchen.
Sounds like part of those arithmetic puzzles. Mostly equivalent to numbers 5 and 6.
Not a cat was in the kitchen.
It sounds a bit literary to begin a sentence with Not a — or with Nary a, for that matter, which means the same thing. This says that there wasn’t even one single cat in the whole kitchen. Compare:
- Not a creature was stirring, not even a mouse.
Kitchen seems unusual here; it is not a big place, after all, so perhaps farm or town would work better here. More natural, but still literary, equivalent formulations might include:
- Not a cat was to be seen, despite the thundering herd of field mice.
- Not a cat was to be seen in the kitchen that day.
- Nary a cat was to be found, the whole hamlet through.