Pryor was the Jets' first-round pick in the 2014 NFL Draft. The 24-year-old safety had 60 tackles, six passes defended and one forced fumble in 15 games last season.
Davis, 28, was a third-round selection by the Jets in the 2012 NFL Draft. He had 338 tackles, eight passes defended, 6.5 sacks, four fumble recoveries and an interception in four seasons for the Jets, before joining the Browns in 2016. Last season he had 99 tackles, two passes defended, two sacks and a forced fumble.
"Calvin is a young-experienced safety that has upside," Browns executive vice president of football operations Sashi Brown said in a team statement. "We are pleased to be able to add him to our defensive back room and just like every player we acquire, we expect him to come in with a hard-working mindset ready to compete. Demario is a guy that we developed the utmost respect for in his time with our team, not only as a professional but also as a person. We appreciate all he did for our organization in his time in Cleveland."
Trade and release rumors swirled around Pryor after the 2017 NFL Draft, when the Jets opted to take two safeties early on. New York snagged LSU safety Jamal Adams with the No. 6 overall pick in the first round and drafted Florida safety Marcus Maye with the No. 39 overall pick in the second round.