Clifton has been the Packers' primary left tackle since 2000, starting 160 games over 12 seasons with quarterbacks Brett Favre and Aaron Rodgers. He started only six games last season because of hamstring and back injuries.
Clifton had offseason back surgery and failed a team physical, sources told ESPN NFL Insider Adam Schefter, who initially reported Clifton would be released.
It's not certain whether Clifton will be able to play again, sources told Schefter.
"Chad is a great Packer and has been an integral part of our success over the past 12 seasons," Packers general manager Ted Thompson said in a statement. "He was a member of some of the most prolific offenses in team history, but more importantly, he is a great person and teammate. We thank Chad for all that he has given the Packers on the field and in the community and wish him and his family well."
Clifton played 165 career games for Green Bay and ranks second in franchise history for the most games played by an offensive tackle behind only Forrest Gregg, who played in 187. Clifton was selected to the Pro Bowl in 2007 and 2010.
Marshall Newhouse played well enough in Clifton's absence last year that it wasn't a sure thing that the veteran would get his job back when he returned near the end of the season. Clifton started the Packers' playoff loss to the New York Giants but was replaced by Newhouse in the third quarter.
Speaking at NFL meetings last month, Packers coach Mike McCarthy told reporters there was "definitely a possibility" that Clifton would stay with the team but said it ultimately would be a medical decision.
"He's someone we'll continue to talk about and get the reports," McCarthy said at the time. "He's down in Tennessee now, so we'll see how he recovers."
With Clifton gone, the Packers' options at left tackle for next season include Newhouse and Derek Sherrod, last year's first-round draft pick. Sherrod is recovering from a broken leg in the Packers' Dec. 18 loss at Kansas City.
The Packers also could consider moving Bryan Bulaga from right tackle.
With the draft set to begin with the first round Thursday night, Clifton's future isn't the only major decision facing the Packers this offseason. There also is uncertainty over the future of safety Nick Collins, who sustained a season-ending neck injury last year. It's not clear if Collins will be able to return.
There also has been speculation that the team could part ways with veteran receiver Donald Driver, although McCarthy said at the league meetings that the Packers "absolutely" still had room for Driver.
"I thought Donald, the second half of the season, he played very well," McCarthy said. "His role, as far as opportunities, was not as high as it's been in the past, but I thought Donald was very consistent down the stretch."
Information from The Associated Press contributed to this report.