Renting in Valencia with a pet cat

@NomadWriter From experience this is what you will find renting as a "foreigner and cat owner" in Valencia, lots of owners don't like foreigners if they are not actively employed here in Spain (the reason is bizarre) then add the pet aspect and your options will become limited, you will most definitely find an apartment however you will have to pay a higher deposit, this could by anywhere from 3-12 months.

Have you decide where in Valencia you would like to live?

Where are you looking?  I suggest idealista as a good starting point, not least because you can filter with a 'admite mascota' option.  As Kevin says, your bigger challenge will be to find a rental based on savings vs regular monthly income

Agree: Idealista frequently lists pet-friendly rentals.  Additionally, I've even seen ads on Idealista that specify a preference for older, stable individuals (some actually say retirees!).

@NomadWriter As others have said, try Idealista.

If you find adverts that don't mention pets at all, then strictly speaking you don't need to mention it either.  But ask for a sample contract, to ensure the contract doesn't mention them either - if the contract doesn't mention them, then you're fine - there's no legal requirement for you to tell the landlord that you have one.

Even if the contract does mention them, if you're dealing direct with the landlord, rather than an agency, it's worth talking to them about it.  Many landlords put "no pets" because they're worried about barking dogs annoying the neighbours, but may accept a docile cat.

BTW - make sure you're familiar with Spain's tenancy law, particularly with regard to deposits, the landlord's right to enter (basically, except in an emergency, the landlord has to right to enter), and "11 month" contracts.