Sowell, 27, started the first six games of last season at left tackle for the Seahawks, but lost his job to George Fant after suffering a knee injury. Sowell later started three games at right tackle before being replaced by Garry Gilliam.
The Bears drafted Kutztown's Jordan Morgan, who played left tackle in college, in the fifth round, but Chicago general manager Ryan Pace hinted that Morgan could line up at guard.
Originally an undrafted free agent, Sowell broke into the league with the Colts in 2012. He then spent three seasons with the Cardinals before signing with the Seahawks on a one-year deal.
Sowell has started 21 games in his NFL career.
ESPN's Josh Weinfuss, Jeff Dickerson and Sheil Kapadia contributed to this report.