Norris agreed to a major-league deal with the Cubs on Wednesday, Jesse Rogers of ESPN.com reports.
Norris was acquired by the Brewers ahead of the 2021 trade deadline but struggled during the second half with a 6.64 ERA, 1.57 WHIP and 18:15 K:BB over 20.1 innings. Overall he had a 6.16 ERA last season and has yet to find consistent success since making the transition from starter to reliever.