Jenks hasn't pitched since June 29 because of soreness on the left side of his upper back.
"We don't want to take a risk," manager Ozzie Guillen said. "We made that decision, and we'll find out after the game [on Wednesday] who is coming up. It's been too many day-by-days."
Guillen said the team wouldn't make a decision on who to bring up until Wednesday.
Jenks hasn't blown a save since early June and had a 1.50 ERA in his last 17 games. He said there have been no major setbacks while he's rested, but the injury continues to nag him.
"I've said it before, there's no reason for me to try and push it," Jenks said. "Where we're sitting right now, if I have to sit the rest of the first half and I'm ready for the second half push, that's more important."
Scott Linebrink earned the save for the White Sox on Sunday. Although reliever Octavio Dotel has been a closer before, Guillen said Chicago could use Linebrink, Matt Thornton or Dotel in the closer role.
Mark Dent is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.