The cost of repairing the ancient bridge and surrounding pathways at Tarr Steps will run into thousands of pounds.
Somerset County Council and Exmoor National Park say the landmark on Exmoor has been affected by landslips, fallen trees and erosion from recent flooding.
Hundreds of people had to be evacuated from homes in Barnstaple after the River Yeo burst its banks.
A woman spent 50 minutes clinging to a branch after being swept away from her flooded car at Umberleigh in North Devon.
Braunton's brand new flood defences were unable to withstand their first test of flooding.