Both pitchers were placed on the 15-day disabled list Friday by the Tampa Bay Rays before the start of their series against the Florida Marlins.
Kazmir, who has a left quadriceps strain, remained with the team. The 39-year-old Percival, sidelined by right shoulder tendinitis, flew home to California to discuss his baseball future with his family.
Percival gave up two runs in one-third of an inning Thursday against Oakland.
"I spoke to him after the game," manager Joe Maddon said. "He felt he was inconsistent in how he felt, and his arm was really bothering him after the game. He was distraught, and he went home to think about things for a little bit. I wouldn't be surprised if he chooses not to come back."
Kazmir is 4-4 with a 7.69 ERA. He said his thigh has bothered him since the Rays' series at Boston two weeks ago.
"I felt something," he said. "I felt I was putting way too much pressure on it, and I've been fighting it ever since. You feel it when you're landing."
The Rays recalled right-hander Dale Thayer and shortstop Reid Brignac from Triple-A Durham, and they'll make another move to call up a starter to take Kazmir's turn Monday at Cleveland. Maddon said he'll go with a closer by committee in Percival's absence.
Percival, a four-time All-Star, ranks eighth on the career saves list with 358. He's 6-for-6 this season but has a 6.35 ERA, and the DL stint is his fourth since the beginning of last year.
"There is only so much you can do when you get to be that age," Maddon said. "He has done everything he could to be a big contributor. He's just not feeling he can be that right now."
Kazmir, who went 12-8 for the AL champions last year, gave up seven runs in 4 1/3 innings in his most recent start Wednesday, a loss to Oakland. Over his past five starts he has allowed 31 runs in 23 innings.
The left-hander said his arm feels fine, and he doesn't anticipate a long layoff. Maddon said the timetable for Kazmir's return is "open-ended, no rush, get it right."
"I've always been concerned that when pitchers start telling me their leg bothers them, there's the potential for the arm to be next," Maddon said. "We just felt it would be wise to back off and wait until it has calmed down."
Kazmir said the injury stems from changes in his motion after he was sidelined with a strained left elbow early last season.
"It was me developing bad habits after I came off the DL in 2008 and protecting my elbow," he said.
Thayer, who will make his major league debut at age 28, had six saves with an 0.93 ERA at Durham. Brignac was hitting .291 with three homers and 15 RBI.