In a message posted on Twitter, Gilmore said he is "currently asymptomatic and will take this as it comes" and urged everyone "to take this seriously."
NFL Network's Tom Pelissero reported Gilmore had an initial positive test on Tuesday and, per NFL protocols, additional tests and a rerun of the initial test confirmed the result.
The Patriots announced on Saturday that one of their players tested positive for COVID-19 but didn't reveal his name. Pelissero and Ian Rapoport reported that same day Cam Newton was placed on the reserve/COVID-19 list.
New England lost to the Kansas City Chiefs 26-10 on Monday night after the NFL shifted the game from Sunday because of positive coronavirus tests from both teams. Chiefs practice squad quarterback Jordan Ta'amu was placed on the reserve/COVID list, per Rapoport and Mike Garafolo.
The game was played after both teams didn't return any additional positive coronavirus tests on Sunday or Monday.
There has been no word at this point if the NFL will attempt to reschedule or postpone the Patriots' home game against the Denver Broncos on Sunday.
Gilmore, the reigning NFL Defensive Player of the Year, played all 56 defensive snaps against the Chiefs. He recorded three solo tackles and forced one fumble.