Any of the items mentioned in the previous paragraph can be used to negotiate. It's the big picture, the more you have to pay for pet fees, utilities and maintenance, the less you can afford to pay for monthly rent. If you can, consider signing a two year lease in exchange for a reduction in monthly rent. I have found that very effective. If not monthly rent, then perhaps in the security deposit or any realtors' fee you are asked to pay.
Finally, if you have pets there are some creative ways to convince a landlord to accept your pet or at least to keep the rent reasonable if they do. For example, I wrote a lease once that said, if the tenants failed to 'clean up' after their dog twice a week the landlord had the right to hire a doggie clean up company and deduct the cost from their security deposit. You can also offer to have the unit professionally cleaned at the end of your time there. If you have your dog professionally groomed, show the receipts to the landlord so they can see you are a responsible pet owner. Same with veterinary records.
In rentals as in all real estate, everything is negotiable.