Meche was concerned after giving up four runs in five innings on Friday night at Pittsburgh and nine runs in 3 1/3 innings in his previous outing against St. Louis.
Guillen was held out of the lineup against the Twins in the series opener.
"His left side is stiff and he needs today off, possibly tomorrow," Hillman said. "Just general stiffness; he's had ongoing at times left-sided stiffness and he's been working inside to get it balanced out. But he's been playing through it and I don't anticipate it being a long time."
Hillman said Guillen's absence was unrelated to the shot he took in the nose from a caroming ball on Saturday night at Pittsburgh.
Guillen was just 5-for-26 (.192) with two doubles on the just-completed six-game trip.
Willie Bloomquist started in right field against the Twins.
Meantime, third baseman Alex Gordon is playing games in Arizona in the hope of returning to the lineup after the All-Star break. Hillman said the reports are about normal for a player returning after hip surgery.
Hillman said that Alex Rodriguez and Mike Lowell, both also coming back from hip surgery, have required time off and so will Gordon when he rejoins the Royals.
"I don't expect him to be 100 percent," Hillman said, "and anybody that does is kidding themselves and is crazy to think you can have a major impact surgery in the middle of the body, and come out and do what these guys do, and be 100 percent in that length of time. It's just not possible."
Shortstop Mike Aviles, rehabbing in Arizona from a sore right forearm, has had some setbacks, according to general manager Dayton Moore.
"We're in the process of making some final evaluations so we can make an accurate decision on what to do next," Moore said. "He's just not progressing very well."
Aviles has been out since May 23; Gordon since April 16.
Center fielder Coco Crisp has been ruled out for the season after undergoing surgery on his right shoulder last week.