Soto, Swarzak and Asche are non-roster invitees who made the club.
Asche, who can provide coverage for the White Sox at third base, first base, designated hitter and the corner outfield spots, earned his way onto the roster by hitting .289 with four home runs and nine RBIs entering Chicago's exhibition finale Saturday.
The 26-year-old is a career .240 hitter with 31 home runs and 125 RBI in 371 games over four seasons with Philadelphia.
"He's a professional hitter," Renteria said. "He's another [player] who has some experience under his belt. He's had some good at-bats over the spring. Obviously, he's been in the big leagues before."
Soto, who played in 78 games for the White Sox in 2015 before spending last season with the Angels, will share catching duties with the left-handed hitting Omar-Narvaez.
With Carlos Rodon beginning the season on the disabled list, either Swarzak or Covey will be Chicago's fifth starter. Because of two early off-days, the White Sox won't likely need a fifth starter until April 17.
Swarzek carries a 4.52 ERA in 217 career games with 32 starts over seven seasons with the Twins, Indians and Yankees. Covey has yet to pitch above Double-A.
"We're still working the details out," Renteria said. "We still have a little bit of time. We've been stretching them out. They've been doing a little extra in the pen after their initial work. Then we'll get to determining exactly what we are going to do with those guys moving forward."