He threw out 31% of attempted base stealers last season (19 of 43) and had a .990 fielding percentage.
In Chicago, Gomes likely will back up starter Willson Contreras.
For his career, Gomes has a .247 average, 117 home runs and 416 RBIs in 10 major league seasons with the Toronto Blue Jays, Cleveland, Washington and Oakland.
The 27-year-old Frazier was released by the Yankees last week.
He was the fifth pick overall in the 2013 amateur draft by Cleveland and was a prized prospect when he was acquired by the Yankees in a July 2016 trade that sent reliever Andrew Miller to the Guardians.
Frazier made his major league debut in July 2017, but the outfielder never has had more than 246 plate appearances in a season. His 2018 season was wrecked when he collided with an outfield wall at Bradenton, Florida, during a spring training game, sustaining a concussion that limited him to 41 plate appearances that season.
He hit .267 with 12 homers and 38 RBI in 2019 but struggled with defense, reaching a low point against Boston that June when he misplayed balls off the bat of Eduardo Núñez, Andrew Benintendi and Michael Chavis.
His defense was much improved by this season, but he hit .186 with five homers and 15 RBI. He didn't play after June 30 because of what the Yankees described as vertigo-like symptoms.
Information from The Associated Press was used in this report.