The Texans cut Banks on Tuesday, Field Yates of ESPN reports.
Banks signed a contract with the Texans on Wednesday.
Banks (leg) is expected to sign with the Texans after working out for the team Tuesday, Aaron Wilson of the Houston Chronicle reports.
Banks signed a one-year contract with the Bears on Saturday, Adam Jahns of the Chicago Sun-Times reports.
Banks hurt his ankle during Sunday's 38-10 loss to the Vikings.
Banks (ankle) was listed as a full participant in Thursday's practice.
Banks (ankle) is inactive Sunday against the Packers, Rich Campbell of the Chicago Tribune reports.
Banks (ankle) is listed as questionable for Sunday's game against the Packers.
Banks (ankle) is inactive Sunday against the Lions.
Banks (coach's decision) is inactive Sunday against the Vikings, Tim Twentyman of the Lions' official site reports.