The University Hospitals of North Midlands said critical care beds at both its sites in Stoke-on-Trent and Stafford are at full capacity.
Whilst this has affected a small number of people, it includes patients with "very serious conditions".
The trust is asking surrounding hospitals to help cope with what it calls unprecedented pressures.
Updates on this story and others in Staffordshire
"Whilst we are all used to working within a very busy environment, instances of both Influenza A and B have put unprecedented pressures on our critical care capacity," Robert Courteney-Harris, trust chief executive said.
"This has blocked out a whole isolation POD and, as these patients require specialist one-to-one nursing care, this has had a significant impact on the available bed capacity."
Surgeons will be keeping in touch with their patients and will reschedule appointments as quickly as possible, the trust added.
Hospital operations have been cancelled in Staffordshire due to an outbreak of flu.