It is thought to have happened on Monday, meaning some departments could not connect to a main server.
Some procedures have been cancelled or have been redirected to other NHS providers as the hospitals try to establish what work can be carried out safely.
The NHS said emergency care continued to be available.
GP services across Bexley, Greenwich, Lewisham, Bromley, Southwark and Lambeth boroughs have also been affected.
A spokesperson from Synnovis said the company had sent in a “taskforce of IT experts” to “fully assess” the impact.
The NHS apologised for the inconvenience and said it was working with the National Cyber Security Centre to understand the impact.
