Kelly signed a two-year, $11.5 million contract with the Cubs on Friday, Jeff Passan of ESPN.com reports.
Kelly started at designated hitter and went 3-for-4 with a home run in Friday's 5-2 win over the Angels.
Kelly will start at catcher and bat eighth in Thursday's game against the Blue Jays.
Kelly is out of the lineup for Sunday's game against the Guardians.
Kelly started at catcher and went 0-for-4 in Sunday's 6-5 extra-inning win over the Twins.
Kelly started at designated hitter and went 2-for-4 with two RBI in Saturday's 5-2 loss to the Twins.
Kelly will start at catcher and bat seventh in Wednesday's game versus the Red Sox.
Kelly went 3-for4 with a two-run homer and an additional run scored during Sunday's 8-7 loss to the Yankees.
Kelly started at catcher and went 1-for-3 in Wednesday's 10-1 loss to the Cardinals.
Kelly was traded to the Rangers on Sunday, Ken Rosenthal of The Athletic reports.