The tenancy agreement (contract) might say pets aren’t allowed. The Consumer Rights Act 2015 prohibits “unfair terms” in a contract. This means a blanket ban on keeping pets in a tenancy agreement might be struck out if challenged in court.
Alternatively, the tenancy agreement might say the landlord’s permission should be sought if the tenant intends to keep a pet. In this case, the landlord’s permission should not be unreasonably refused. What amounts to a reasonable refusal will vary with the circumstances. For example, it might be reasonable to refuse permission to keep a large dog in a small flat.
The tenancy agreement might not mention pets, in which case it will likely be harder for landlords to argue that pets are not allowed.
Landlords cannot charge a fee for agreeing to a request to keep a pet and cannot ask for a higher deposit if this would breach the deposit cap requirements in the Tenant Fees Act 2019 (see page 41 of Tenant Fees Act – Tenant Guidance.pdf).