Kessler spent time on New England's practice squad last season. He was inactive for 11 regular-season games as the third-string quarterback behind Brady and Jarrett Stidham.
The Patriots agreed to a one-year deal with former Brady backup Brian Hoyer earlier this month.
Kessler was selected by Cleveland in the third round of the 2016 draft. He has played in 17 regular-season games with 12 starts during his four-year career that also included a stop in Jacksonville in 2018. He has completed 224 of 349 passes for 2,215 yards, eight touchdowns and five interceptions.
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