The new deal, reportedly worth a total of $144 million over seven years, would make Garrett the highest-paid non-quarterback in the NFL.
Garrett, 24, was set to enter the final two years of his rookie contract, after the Browns opted to pick up the fifth-year option of his rookie deal.
Per the report, Garrett is expected to sign the deal, according to the NFL Network.
Garrett missed the final six games of the 2019 season due to suspension stemming from an on-field altercation with Pittsburgh Steelers quarterback Mason Rudolph, but the league lifted his suspension in February.
Garrett has amassed 30.5 sacks, 65 quarterback hits, 104 tackles and 32 tackles for loss over 37 career games.