Colome has 96 saves in six seasons with Tampa Bay and Seattle, earned $5.3 million this year and is eligible for salary arbitration. The 29-year-old right-hander was an All-Star in 2016 and became the first Rays pitcher to lead the league in saves in 2017 with 47. He was dealt to the Mariners last May.
Colome was used as a set-up man by the Mariners and had a 5-0 record and one save with a 2.53 ERA in 47 appearances. .
Narvaez, 26, batted .275 with nine homers and 30 RBIs in 97 games for the White Sox and will make the league minimum in 2019.
"Omar has quietly developed into one of the tougher outs among catchers in MLB, his contact and on-base skills are welcome additions to our lineup," general manager Jerry Dipoto said in statement. "The fact that he is also under club control through 2022 made him a great fit for our club."
Also Friday, the White Sox declined to offer contracts to outfielder Avisail Garcia and Matt Davidson, allowing the pair to become free agents. And they agreed to a $1.55 million, one-year deal with outfielder Leury Garcia.
The Associated Press contributed to this report.