Bobby Wagner has the most games with 10+ assists, with 3 games.

most 10 assist games
Interpreted as:
most 10 or more assist regular-season games by a player
WagnerKuechlyBowmanGreenwayThomasBruschiDavisCampbellBrownFranklinMosleyAlexanderOgletreeHightowerMcKinneyChancellorTrevathanJacksonDavidSmithIrvingNinkovichHenderson32222211111111111111111
NAMEGPASTTKLCOMBSCKSCKYINTINT YDSINT TDFFFRFR TD
Bobby WagnerBobby Wagner33414481.09000000
Luke KuechlyLuke Kuechly22517420.00100000
NaVorro BowmanNaVorro Bowman22311340.00000000
Chad GreenwayChad Greenway2239320.00000000
Zach ThomasZach Thomas22315380.00000200
Tedy BruschiTedy Bruschi2208280.00000000
Todd DavisTodd Davis1105150.00000000
De'Vondre CampbellDe'Vondre Campbell1107170.00000200
Preston BrownPreston Brown1104140.00000000
Zaire FranklinZaire Franklin1101110.00000000
C.J. MosleyC.J. Mosley1102120.00000000
Kwon AlexanderKwon Alexander11011210.00000100
Alec OgletreeAlec Ogletree1104140.00000000
Dont'a HightowerDont'a Hightower1103131.512.5000000
Benardrick McKinneyBenardrick McKinney1106161.06000000
Kam ChancellorKam Chancellor1125170.00000000
Danny TrevathanDanny Trevathan1136190.00000000
D'Qwell JacksonD'Qwell Jackson1103130.00000000
Lavonte DavidLavonte David1123150.00000000
Daryl SmithDaryl Smith1103130.00000110
Nate IrvingNate Irving1112130.00000000
Rob NinkovichRob Ninkovich1130130.00000000
Erin HendersonErin Henderson1104140.00000000
Paul WorrilowPaul Worrilow1109190.00000000
A.J. HawkA.J. Hawk1102120.00000000
StatMuse has game-level data for assists going back to the 2001 season.