Head coach Vance Joseph and president of football operations/general manager John Elway met with Lynch on Sunday to break the news to him.
"Vance and I met with Paxton this morning and informed him that it's best for everyone to make this move. We appreciate all of Paxton's hard work as a Bronco, and we wish him well in the future," Elway said in a statement.
Broncos wide receiver Emmanuel Sanders said on NFL Network on Sunday night that a change might be good for Lynch.
"I'm sad to see him leave, but I'm happy for him because he needs the change of scenery. I remember that second preseason game, we were playing against the Bears. He goes in and the crowd is just booing him. It does something to you as a player," Sanders said.
Elway said Saturday night that part of the thinking was Kelly's inexperience. He was a seventh-round pick in 2017 who spent his rookie season on injured reserve, and that played a part in the decision to retain Lynch.
"Every decision we make, we try to make the best decision for the Broncos, and sitting right now, where we are at quarterback and who knows our system ... I thought at this point in time that's the best decision for us," Elway said Saturday. "Paxton continued to improve, but he is the third-string quarterback. Looking at it, looking at the situation, with Chad not having a lot of experience and not having played a regular-season game before, we felt like we had to go with three quarterbacks, and Paxton was our third one."
Lynch, whom the Broncos traded up to select in the first round of the 2016 draft (26th overall), lost back-to-back training camp battles with Trevor Siemian the previous two summers. This summer, Lynch slid to No. 3 on the depth chart after the Broncos' preseason opener and wasn't able to regain the backup spot.
Lynch did have his strongest outing of the preseason Thursday night against the Arizona Cardinals, completing 14 of 15 passes for 128 yards and two touchdowns just two days before the Broncos made the bulk of their cuts.
ESPN's Jeff Legwold contributed to this report.