The news outlets, citing unnamed team sources, reported the two-time Pro Bowl choice will have the procedure sometime in June, which would likely keep him from starting training camp on time.
Gronkowski, 24, has also undergone four operations on his left forearm since November, with the last operation May 20.
The tight end originally sustained the injury in mid-November, then broke the forearm again in the AFC divisional playoff round Jan. 13 against the Houston Texans.
The latest surgery was meant to replace the plate in the forearm and clear out an infection. The Patriots expected an eight- to 10-week rehabilitation period for that procedure.
Gronkowski hauled down 11 touchdown catches last season, finishing the campaign with 55 receptions for 790 yards. In his three-year career, he has made 187 grabs for 2,663 yards and 38 touchdowns.