Bunkley was injured during the Eagles' 27-24 victory Sunday night at San Francisco, and Reid revealed the extent of the lineman's injury on his radio show on WIP-AM.
"He had ligament damage in there," the coach said, according to the Eagles' official website. "The nerve to where your funny bone is, that nerve shifted, and so they have to go in and straighten it out. It's not a career-ending thing, but it's going to take time to get him back."
Bunkley was the 14th overall pick in the first round of the 2006 draft. He had nine tackles this season.
Peters was expected to undergo an MRI exam on his sprained left knee Monday, according to the Philadelphia Inquirer.
"He's got possibly some cartilage damage in there," Reid said. "He can't support his weight on it, so it collapses on him."
The Associated Press contributed to this report.