Delmonico, 26, had been sidelined by season-ending shoulder surgery in late May. He hit .227 with a .709 OPS in 152 games across parts of three seasons with the White Sox.
In other moves, Chicago placed lefty reliever Jace Fry on the 10-day injured list and purchased the contract of right-hander Odrisamer Despaigne.
Fry has left shoulder soreness, and his move to the IL is retroactive to Saturday. He has a 5.79 ERA in a team-high 28 appearances this season.
Despaigne, 32, is scheduled to start when the White Sox host the Nationals on Monday evening. He has a 4.94 ERA across parts of five MLB seasons. The White Sox signed him on May 19, several days after the Reds released him.