A source told ESPN that Hayward is likely to need surgery to repair the injury.
Hayward, 33, signed with the Falcons, his hometown team, in March after playing six years in the AFC West -- five with the Chargers and last season with the Raiders. He immediately stepped into the starting lineup and had 17 tackles and an interception in his first six games in Atlanta.
The Falcons will likely replace him in the starting lineup with second-year pro Darren Hall, whom the team drafted out of San Diego State in the fourth round in 2021.
"Still early in the week there as we put the plan together, but we're confident in our guys," coach Arthur Smith said Monday. "Guys have stepped up and played really well and Darren did. Darren's had a good camp, his number was called [Sunday], made a huge play on that deep ball to [Brandon] Aiyuk."
The 22-year-old Hall had two tackles and two pass breakups against San Francisco. He has played 20 games over his career, including one start.
Starting cornerback A.J. Terrell and nickel corner Isaiah Oliver also left Sunday's win over the 49ers with injuries. Smith said Monday that the only one he was concerned about being long term was Hayward's.
To replace Hayward on the roster, the Falcons signed tight end MyCole Pruitt, who had been on the practice squad and caught a touchdown pass in Sunday's win. The team also brought back inside linebacker Dorian Etheridge and defensive lineman Jalen Dalton to the practice squad. Both players were with the team in camp but were waived with an injury designation.