Zappe (coach's decision) is inactive but will serve as Cleveland's emergency third quarterback.
Cleveland signed Zappe to its active roster Wednesday.
The Browns will sign Zappe off the Chiefs' practice squad Tuesday, Tom Pelissero of NFL Network reports.
The Chiefs signed Zappe to their practice squad Wednesday, Jordan Schultz of BleacherReport.com reports.
The Patriots are slated to waive Zappe, Mike Reiss of ESPN reports.
Zappe completed 12 of his 20 pass attempts for 108 yards and ran once for six yards in Thursday's 17-3 preseason win over the Panthers.
With veteran newcomer Jacoby Brissett and 2024 first-rounder Drake Maye viewed as roster locks, Zappe figures to compete with rookie sixth-rounder Joe Milton and Nathan Rourke for slotting on the Patriots' depth chart, Andrew Callahan of the Boston Herald reports.
Zappe appears set to operate as the No. 2 quarterback behind Jacoby Brissett, who is slated to sign a one-year, $8 million contract with the Patriots, John Keim of ESPN reports.
Zappe completed 16 of 26 passes for 209 yards and three interceptions while adding 37 rushing yards and a touchdown on five carries in Sunday's 27-21 loss to the Bills.
Zappe implied Tuesday that he'll remain the Patriots' starting quarterback Thursday at Pittsburgh, Andrew Callahan of the Boston Herald reports. "Coach (Bill Belichick) has made it pretty clear, but I'll let him announce it to everybody," Zappe said.