General manager Kevin Towers said in a statement Monday night that the team planned to continue negotiations with Hudson but was unable to reach an agreement before Monday's deadline. The non-tender move means he can negotiate with other teams.
Hudson is recovering from a second Tommy John elbow surgery. He has not pitched since 2012. During rehabilitation for his first surgery, he re-injured the elbow in June, requiring a second operation.
The Diamondbacks did tender offers to their other four arbitration-eligible players -- outfielder Gerardo Parra, right-handed pitchers Josh Collmenter and Brad Ziegler and left-handed reliever Joe Thatcher.