Huber was a fifth-round draft pick of the Bengals in 2009.
He holds the franchise record for gross average (45.0) and net (39.8) punting yards.
So far this offseason, the Bengals have acquired offensive tackle Cordy Glenn from the Bills and signed offensive lineman Bobby Hart and defensive lineman Chris Baker while declining to pick up a contract option for cornerback Adam Jones.
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They have also seen quarterback AJ McCarron, defensive lineman Chris Smith and offensive lineman Andre Smith sign elsewhere.