Eyre experienced tightness in his elbow after his last outing on Saturday against the White Sox, when he gave up three runs on four hits over one inning.
The candidates include Carmen Pignatiello, who has not given up a run or walked a batter this spring; Sean Marshall, who will be tested in relief this week; and Les Walrond. Marshall, who was an option in the rotation, has had success against left-handers.
Pignatiello, called up last year, has had success with his sinker.
"It's been a really good pitch for me this spring," Pignatiello said, "and I'm getting a lot of ground balls with it, and I feel that's important pitching at Wrigley Field where you've got the wind and the thick grass and if you can get a lot of ground balls, you'll be successful."
The left-hander grew up in Chicago, and was the Cubs' 20th-round pick in the 2000 First-Year Player Draft. He lives in Aurora, Ill., now.
"This is my eighth year, and I'm fortunate to get the opportunity to play for the team I grew up rooting for, and hopefully, I can have a long career playing for them," he said.
This is a tough week for players on the bubble. Cubs manager Lou Piniella has said that he wants to finalize the roster before the Cubs leave for Las Vegas.
"There's nothing you can do about it," Pignatiello said. "You have to go out and do your job. If you do your job, you've got a chance."
The bench also has yet to be finalized. Piniella said he didn't like keeping people in limbo. Henry Blanco and Daryle Ward will be on the bench, but the other spots have yet to be determined.
"Any combination we take, I'll be pleased with, because I like all these kids," Piniella said.
Infielder Alex Cintron has been slowed because of a sore left hamstring, but did play shortstop on Monday. Eric Patterson has played well all spring, getting time at both second base and in the outfield, but Piniella said he was "on the outskirts."
Carrie Muskat is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.