"As soon as possible," Chester told ESPN.com of the timetable for the surgery. "The rotator cuff kind of got jammed up a little bit. It's nothing major, but it's an issue that's going to have to get addressed."
Chester suffered the injury in a Week 12 loss to Minnesota and played through the pain. He was held out of Wednesday practices from Week 13 on.
Chester signed a one-year, $2.8 million contract with the Falcons and will hit free agency in March if not re-signed. He turns 33 next Tuesday and wants to continue his playing career in Atlanta.
"I've really enjoyed my time here," Chester said. "Things didn't go the way we would have liked them to go, but I really enjoyed myself here. Coach [Dan] Quinn, the coaches, the owner are great.
"There are a lot of moving parts. I would love to come back here. The nature of the business is pretty complicated, but I love it here."
Chester was asked how many more years he wants to play.
"I don't know, I'm week to week," he said. "I'm day to day, really. I've still got it, but there's a lot ... I've got to think about my family and my body and my health and stuff. There's a lot to think about."
The Falcons have plenty to consider as well. Quinn said solidifying the lines on both sides of the ball is at the forefront of his thoughts and a "huge emphasis" this offseason. The offensive line didn't perform up to expectations, although there were moments the front five looked good paving holes for 1,000-yard rusher Devonta Freeman in the zone blocking scheme.
Chester was known for his ability to run block in offensive coordinator Kyle Shanahan's system, which he became familiar with while playing under Shanahan in Washington. However, Chester is viewed as undersized in terms of pass protection.
Quinn was asked to evaluate the performances of his veteran starting guards, Chester and Andy Levitre.
"I thought both of them, in the run game, they showed their quickness to get outside," Quinn said. "As we're going through the whole process of it, both of them have unique stuff in terms of the speed that they can play with. I thought Andy, especially, adapted quickly to the scheme, where Chris had been part of the wide-zone scheme for a while with Baltimore and at Washington.
"I was pleased with both of them."
That doesn't necessarily mean they'll be the starting guards next season. Quinn and his coaches are still in the midst of the roster evaluation process and seem likely to move on from Chester and Levitre. The Falcons need to get bigger on the interior of the offensive line, and Chester and Levitre were more stop-gap players than anything else. Chester fought hard while playing with one arm, while Levitre wore down at the end of the season and finished with a team-high 11 penalties.