The slaughter is being carried out as a precautionary measure to prevent the virus spreading, the Department for Environment, Food and Rural Affairs (DEFRA) said.
The farms are about two miles (3.2 kilometres) from the farm where the H7N3 strain was first confirmed earlier this week.
Although harmful to poultry, H7N3 is much less of a threat to humans than the H5N1 form of the virus, which was confirmed in a dead swan in Scotland in early April, the only case so far in Britain.
Public health officials said Sunday that three more poultry workers have contracted conjunctivitis in addition to one who developed the eye condition and was tested positive for H7N3 Friday.