Gesicki is in line to sign a one-year deal with the Bengals, Adam Schefter of ESPN reports.
Gesicki finished the 2023 season with 29 catches on 45 targets for 244 yards and two TDs in 17 games.
Gesicki caught four of six targets for 35 yards in Sunday's 27-21 loss to the Bills.
Gesicki is expected to serve as the Patriots' top tight end Sunday in Buffalo with Hunter Henry (knee) inactive for the contest, Chris Brown of the Bills Radio Network reports.
Gesicki, who was on the field for 38 of 58 snaps on offense in Sunday's 26-23 win over the Broncos, caught both of his targets for 15 yards and a touchdown.
Gesicki, who was on the field for 21 of a possible 53 snaps on offense, per Mike Reiss of ESPN.com, caught one of his two targets for seven yards in Sunday's 27-17 loss to Kansas City.
Gesicki was on the field for 19 of a possible 57 snaps on offense in Thursday's 21-18 win over the Steelers, Mike Reiss of ESPN.com reports.
Gesicki logged seven of a possible 64 snaps on offense in Sunday's 6-0 loss to the Chargers, Mike Reiss of ESPN.com reports.
Gesicki did not record a catch on two targets during Sunday's 10-7 loss to the Giants.
Gesicki was on the field for 46 of a possible 69 snaps in Sunday's 10-6 loss to the Colts, Mike Reiss of ESPN.com reports.