Broad has gone under the knife in a bid to stop persistent knee problems and the England and Wales Cricket Board (ECB) are hopeful he will recover in time for next year's Cricket World Cup.
The 28-year-old took 19 wickets in England's 3-1 Test series win over India in July and August but has been absent throughout the subsequent one-day series.
Prior has not played since the second Test against India in July due to his injury.
An ECB statement read: "The ECB medical team today confirmed that England and Nottinghamshire bowler Stuart Broad has undergone an operation on his right patella tendon.
"Broad will begin a rehabilitation and recovery programme ahead of the tri-series against Australia and India early next year.
"England and Sussex wicket-keeper 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."