Berry reported that a source said "the Giants don't believe [Engram] can ever stay healthy."
The 25-year-old has yet to play a full season since New York drafted him in the first round of the 2017 NFL draft, missing 14 combined games.
Engram was on pace for a career year in 2019, having caught 44 passes for 467 yards and three touchdowns through eight games, before a mid-foot sprain took him out of action and eventually landed him on injured reserve.
The same reason the Giants might be looking to offload Engram will likely make trading him difficult.
The New York Post's Ryan Dunleavy reported he's in a walking boot after undergoing surgery in December and could be sidelined until training camp.
Engram's contract situation complicates matters further.
The former Ole Miss star is entering the final guaranteed year of his rookie deal and has a fifth-year option for 2021. New York has until May 30 to either pick up that option or decline it, thus making Engram a free agent next spring.
NFL.com's Gil Brandt argued the Giants should keep him for 2021, but other teams might be willing to gamble if they can wait a year and sign Engram rather than giving up a draft pick or another trade asset to land him now.