Broad, who was the second-highest wicket-taker behind James Anderson in the five Tests against India, underwent a knee operation and is expected to join the England ODI squad in January next year.
Broad, 28, had decided in the middle of the Test series that he will get his knee problem, which was giving him trouble for long, sorted but the all-rounder also had to take a hit on the nose in the fourth Test.
Meanwhile Prior, 32, who sat out most of the season with a left Achilles problem, decided to get over it after the team's 95-run loss at Lord's in the third Test.
Thursday evening's press release read: "The ECB medical team today confirmed that England and Nottinghamshire bowler Stuart Broad has undergone an operation on his right patella tendon.
"England and Sussex wicketkeeper Matt Prior has undergone surgery on his left Achilles and will begin a rehabilitation and recovery programme with England and Sussex over the winter period.