Turner left Friday's practice early, and Saints coach Dennis Allen said after practice that the team was still determining the severity of the injury.
The New Orleans native was expected to compete for a spot on the team after signing with the Saints on Tuesday. Turner was with the second-team offense when the Saints ran 11-on-11 drills Thursday.
The 30-year-old was a 2014 third-round pick of the Carolina Panthers. He spent the previous season with the Washington Commanders, playing in 16 games with 12 starts. He also spent time with the Los Angeles Chargers and the Pittsburgh Steelers.
"I think it was just one of my goals to play as long as I can," Turner said when asked if he always had the goal of returning to New Orleans. "I just turned 30 and I'm still playing a kid's game, so I'm just appreciative of the opportunity. I'm a decade in, so no complaints on my end."
Saints offensive tackle Trevor Penning returned to practice Friday after missing Thursday because of a minor foot injury, and offensive linemen Ryan Ramczyk and Cesar Ruiz had a planned rest day and did not practice.