Kwiatkowski has served as Washington's defensive coordinator since 2014 and also spent the 2020 season as the team's linebacker coach.
The 55-year-old previously spent time as a defensive assistant with Eastern Washington and worked under former Huskies head coach Chris Petersen at Boise State - Kwiatkowski's alma mater - from 2010-13.
Kwiatkowski split Washington's defensive coordinator duties with current Huskies head coach Jimmy Lake in 2016. The program led the Pac-12 in scoring and total defense from 2015-18.
The reported hiring marks another significant addition to new Longhorns head coach Steve Sarkisian's growing staff. Sarkisian already signed Jeff Banks from Alabama to serve as Texas' new special teams coach.
Texas' defense allowed 28.5 points and 406.6 yards per contest in 2020.