Bannister, also out with right shoulder fatigue, will seek a second opinion.
"Banny is going to New York tomorrow -- they've made a diagnosis here and, just as many athletes do, he just wants to get a second opinion," Hillman said.
He declined to elaborate on the diagnosis on Bannister, who last pitched at Oakland on Sept. 2, when he left the game in the second inning.
Meche's spot in the rotation has been taken by Robinson Tejeda, who's put together 11 1/3 shutout innings in his two starts. Tejeda will next pitch on Tuesday night at Detroit.
Bruce Chen filled in for Bannister on Tuesday against Detroit and pitched 4 2/3 innings in a Royals victory. However, Bannister's spot will be taken next by left-hander Lenny DiNardo, who started on Thursday against the Tigers.
Chen will return to bullpen duty.
Meche will continue to undergo treatment and the Royals believe he might be able to at least resume playing catch later this week.
If his season is over, Meche will finish with a 6-10 record and 5.09 ERA in 23 starts and 129 innings. That's a big comedown from 2008, when he went 14-11 with a 3.98 ERA and made 34 starts, the same number he made in his first season with the Royals. For part of the season, he was troubled by back problems.
Bannister stands 7-12 with a 4.73 ERA in 26 starts. He has not won since Aug. 2 at Tampa Bay; since then, he's 0-5 in six starts.
Dick Kaegel is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.