Since 2000, the Green Bay Packers have the most passing touchdowns by a team, with 680 touchdowns.

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which nfl team has the most td passes since 2000
PackersSaintsColtsPatriotsChargersEaglesCowboysSeahawksSteelersBroncosBuccaneersChiefsFalconsVikingsGiantsBengalsLionsRamsCardinals49ersTitansRaidersRavens680667634624586545543541540532521517511505502491489478465459457455447
TEAMGPTDCMPATTPCTYDSAVGYDS/GTD%INTINT%SCKSCKYRATE
Green Bay PackersGreen Bay Packers3486807,84212,35463.591,2217.4262.15.52742.27425,01794.9
New Orleans SaintsNew Orleans Saints3476678,37112,78565.595,8637.5276.35.23082.46314,13895.2
Indianapolis ColtsIndianapolis Colts3486347,91412,51063.391,2727.3262.35.13022.45733,66592.0
New England PatriotsNew England Patriots3486247,81512,26563.790,4667.4260.05.12361.96844,24094.9
Los Angeles ChargersLos Angeles Chargers3475867,50711,87663.288,4497.4254.94.93272.86894,36390.8
Philadelphia EaglesPhiladelphia Eagles3485457,49612,39860.586,9777.0249.94.43062.58575,33986.1
Dallas CowboysDallas Cowboys3475437,25411,54762.884,9717.4244.94.73413.07805,04388.5
Seattle SeahawksSeattle Seahawks3475416,84711,03562.079,9827.2230.54.92842.69115,76189.6
Pittsburgh SteelersPittsburgh Steelers3475407,26811,57662.885,7907.4247.24.73212.87835,04089.3
Denver BroncosDenver Broncos3475327,29311,88261.485,0197.2245.04.53322.86994,51586.3
Tampa Bay BuccaneersTampa Bay Buccaneers3475217,34012,05260.984,3857.0243.24.33603.07824,90284.0
Kansas City ChiefsKansas City Chiefs3475177,19911,58762.184,3587.3243.14.52742.47684,60989.2
Atlanta FalconsAtlanta Falcons3475117,37111,84562.284,9377.2244.84.33082.68225,24187.4
Minnesota VikingsMinnesota Vikings3475057,21311,29463.981,5187.2234.94.53112.88755,33788.8
New York GiantsNew York Giants3475027,32212,16260.284,1216.9242.44.13472.97054,75682.9
Cincinnati BengalsCincinnati Bengals3474917,17211,74561.179,4136.8228.94.23422.97665,00082.9
Detroit LionsDetroit Lions3474897,65412,76260.085,6636.7246.93.83813.08725,59480.4
Los Angeles RamsLos Angeles Rams3474787,55212,22161.887,0357.1250.83.93743.18545,91083.5
Arizona CardinalsArizona Cardinals3474657,39312,37159.884,4596.8243.43.83913.28545,89979.7
San Francisco 49ersSan Francisco 49ers3474596,61810,93860.576,6517.0220.94.22952.78935,58384.5
Tennessee TitansTennessee Titans3484576,66810,95960.877,5757.1222.94.23122.87324,74984.3
Las Vegas RaidersLas Vegas Raiders3474557,11811,77560.581,0866.9233.73.93162.77874,87282.8
Baltimore RavensBaltimore Ravens3474476,95111,48660.575,8656.6218.63.93162.87585,03781.5
Washington Football TeamWashington Football Team3474417,06811,56861.180,2606.9231.33.83192.88325,84683.1
Buffalo BillsBuffalo Bills3474406,68811,08360.375,2256.8216.84.03313.08825,60481.4
Miami DolphinsMiami Dolphins3484377,01911,58160.677,7506.7223.43.83533.08445,70980.4
Chicago BearsChicago Bears3474346,79711,37459.874,1806.5213.83.83693.28395,46878.3
Carolina PanthersCarolina Panthers3484336,62311,21159.177,2346.9221.93.93643.28175,82179.4
Jacksonville JaguarsJacksonville Jaguars3474326,90311,60159.577,1926.7222.53.73032.69195,61380.9
Houston TexansHouston Texans3154216,46610,33862.574,1847.2235.54.12872.88265,28586.1
New York JetsNew York Jets3474066,59011,08759.473,8836.7212.93.73813.48095,16077.3
Cleveland BrownsCleveland Browns3483966,66411,45558.275,0826.6215.83.53923.48795,75575.1
StatMuse has team, season-level data for passing touchdowns going back to the 1921 season.