Multiple teams reportedly placed waivers claims for Willis on Wednesday, but the Jets had the highest waiver priority among these teams.
Reports from earlier in the day mentioned that Enunwa suffered a season-ending neck injury, so this move was expected.
Willis, 24, is a former third-round pick of the Bengals back in 2017. Cincinnati placed him on waivers on Tuesday.
Willis is currently in the third year of his four-year, $3.61 million contract and set to make a base salary of $754,750 for the 2019 season.
In 2018, Willis appeared in all 16 games for the Bengals and recorded 20 tackles, one sack, a fumble recovery and a pass defense.
Enunwa, 27, is a former sixth-round pick of the Jets back in 2014. He played out the final year of his three-year, $1.995 million and made a base salary of $615,000 for the 2017 season before returning to the Jets on a one-year restricted deal worth $2.914 million.
The Jets re-signed him to a four-year, $36 million deal which included $20 million guaranteed.
In 2019, Enunwa appeared in one game for the Jets and caught one pass for -4 yards receiving.