NFL Network earlier reported the extent of and timetable for Smith's injury. Team sources told ESPN that they are "optimistic" that it won't be a long-term injury.
Smith was injured at practice Tuesday morning when he got rolled up on during a team period.
"One of the defensive guys threw themselves into him, or got thrown into him," said right tackle Demar Dotson. "I don't know what happened, but we've just got to do a good job of staying up. When it gets late in training camp like this, people get tired and sloppy. That's kind of what happened."
Smith shouted in pain while being tended to by team trainers, and eventually walked off the field with some assistance.
"That's a guy that we need," Dotson said. "He means everything [to this team]. We don't have [anybody] that can play left tackle the way he can play left tackle."
Starting safety Justin Evans also suffered a right ankle injury that forced him to leave practice, but the team does not believe it's serious.
Smith is entering the fourth year of his career and has yet to miss a start.
"That's probably one of Donovan's best characteristics," head coach Dirk Koetter said. "He's very reliable and he's been very durable over the course of his career."
Since entering the league in 2015, Smith has played the seventh-most snaps in the NFL (3,088) according to ESPN Stats & Information. He also has started all 48 games in his career -- tied for the fourth-longest active streak of consecutive starts by an offensive tackle.