Nagy, 39, replaces John Fox, who was fired after posting a 14-34 record in three seasons.
The Bears interviewed five candidates before meeting with Nagy on Sunday, the day after the Chiefs' playoff loss to the Tennessee Titans.
Nagy spent the last 10 seasons coaching under Andy Reid in Philadelphia and Kansas City. Reid promoted Nagy to Chiefs offensive coordinator this year -- Nagy served as Kansas City's co-offensive coordinator in 2016 -- and officially relinquished playcalling duties to the young assistant late in 2017. Kansas City won four in a row to close out the regular season before Saturday's collapse against the Titans.
Nagy joins Marc Trestman (2013-14) as the only Bears head coach since the merger to have previously worked as an offensive coordinator.
Since 2015, the Bears' offense is ranked tied for 30th in points per game (18.3).
Prior to 2016, Nagy spent three seasons as Kansas City's quarterbacks coach and worked closely with veteran Alex Smith, who made his first Pro Bowl in 2013.
Nagy will be charged with developing Bears quarterback Mitchell Trubisky, whom general manager Ryan Pace moved up one spot to draft second overall last spring. Pairing Trubisky with the correct offensive mind was a major point of emphasis when Pace embarked on the coaching search.
Before settling on Nagy, the Bears interviewed New England Patriots offensive coordinator Josh McDaniels, Philadelphia Eagles quarterbacks coach John DeFilippo, Minnesota Vikings offensive coordinator Pat Shurmur, Vikings defensive coordinator George Edwards and Bears defensive coordinator Vic Fangio.
According to multiple sources, Nagy thought highly of Trubisky leading up to last year's NFL draft. That fact no doubt worked in Nagy's favor during the interview process.
Nagy will be the play-caller for Trubisky, sources told ESPN's Chris Mortensen. The Bears would like to keep Fangio, but are aware their defensive coordinator has other options.
Before joining the Chiefs, Nagy coached for three years under Reid with the Eagles, where he served two seasons as Philadelphia's quality control coach (2011-12). Nagy held the title of coaching assistant for the Eagles in 2010. Nagy was a coaching intern for Reid in 2008-09.
Nagy becomes the sixth NFL head coach to have previously served as an assistant under Reid. Ron Rivera, Doug Pederson, Todd Bowles, John Harbaugh and Sean McDermott are the others.
A former quarterback for the Arena Football League, Nagy played collegiately at Delaware.
Nagy inherits a team that finished 5-11 in 2017. The Bears haven't reached the playoffs since 2010, and have missed the postseason in 10 of the last 11 years. The Bears have finished last in the NFC North four straight seasons.
The Bears will introduce Nagy at a news conference on Tuesday.