Addenbrooke's Hospital in Cambridge said two wards were now shut after flu affected dozens of patients. Earlier three wards and four treatment areas had been closed by the outbreak.
Flu has also hit public areas and people are urged to stay away from A&E.
Routine operations are being delayed and care services are affected by flu, medical director Dr Jag Ahluwalia said.
"The hospital is full and we're struggling to cope. Some patients have had to be diverted to other hospitals," he added.
"We have high numbers of people with flu across the hospital.
"For coughs, colds and flu-like illnesses, people are better off at home, resting and drinking plenty of fluids.
"We need people to stay away from the hospital and ask themselves - do you need to be here?"
Influenza is a virus so cannot be treated with antibiotics and in most cases it will run its course, the hospital said.
Urgent operations including surgery for cancer patients are being delayed by an outbreak of flu at a hospital.