The councils, which share IT systems, are working with specialist cyber incident experts and the National Cyber Security Centre (NCSC) to protect data and restore services.
A spokesperson for the Royal Borough of Kensington and Chelsea stated it is currently 'too early' to determine if any data has been compromised or who is behind the attack.
The incident has affected various systems, including phone lines, and residents may experience delays in responses and services.
The Information Commissioners' Office has been informed, and the NCSC is aware of the incident and is working to understand its potential impact.