The Royals recalled right-hander Yohan Pino from Triple-A Omaha before playing Oakland.
Holland has four saves this season and has not allowed a hit in four scoreless innings. He has been an All-Star the last two years.
Holland called it a "slight stretch in my pec area."
"I tried to pitch through it," Holland said. "The ability to recover each and every day as a reliever is important. For me at this point of the season it's smarter for me to go on the DL. I just didn't feel comfortable."
Yost said Holland felt the problem after a save against the Angels on April 11.
"It just didn't feel good and didn't get any better after I was done throwing," Holland said. "I'm pretty optimistic. I feel a lot better today, but I'm a little sore."
"But to help this team out, I've got to be able to throw every day, multiple days in a row. I'm in a situation where I can't do that right now. Luckily, we've got a real deep bullpen."
Yost said "it had been five days and he wanted to give it a try last night. Afterward, you could tell he was cutting off his pitches and it was still bothering him."
"We felt the best course is shut him down for a week and see if can get this inflammation out of him," he said.
Yost said the club is still waiting MRI results.
"We have to determine the extent of it and that will determine how long the shutdown will be," Yost said. "Hopefully not too much longer after the 15 days."
Holland had 46 saves last year in the regular season, then earned seven more in the postseason for the AL champions.
Bullpen depth is a strength for the Royals.
"We've got three guys on this club that can close for any team," Yost said. "We've got enough candidates in house to be able to handle it."
The 31-year-old Pino pitched seven scoreless innings and allowed one hit with no walks and six strikeouts last Sunday at New Orleans.
Pino was 2-5 with a 5.07 ERA in 11 starts last season with the Minnesota Twins.
The Royals also recalled Orlando Calixte from Triple-A to provide some depth in the infield with shortstop Alicedes Escobar's status uncertain. Escobar sprained his left knee when he was slid into by Oakland's Brett Lawrie on Friday night. Outfielder Reymond Fuentes was optioned back to Omaha to clear the roster space.