Bengals coach Zac Taylor announced the signing and also told reporters that the team has released defensive tackle Geno Atkins, offensive tackle Bobby Hart and backup quarterback Ryan Finley.
Hart started 13 games last season for the Bengals. He started all 16 games on the Bengals offensive line in 2018 and 2019.
Taylor said Reiff will play right tackle and Jonah Williams will play at left tackle next season. Reiff started 15 games for the Vikings in each of the last two seasons. The 6-foot-6, 305-pound offensive lineman has appeared in 135 games during his nine-year NFL career.
Reiff entered the league as the No. 23 overall pick by the Detroit Lions in the 2012 NFL Draft. He spent the first five seasons of his career with the Lions.
The Bengals had the third-worst offensive line in the NFL last season, according to Pro Football Focus.
Burrow, the No. 1 overall pick in the 2020 NFL Draft, sustained a season-ending knee injury when he was hit by defenders in Week 10 of last season. He was sacked 32 times in the first 10 starts of his NFL career, including eight times in a Sept. 27 game against the Philadelphia Eagles.
Atkins, an eight-time Pro Bowl selection and two-time All-Pro, had spent his entire 11-year career with the Bengals. Finley appeared in eight games and had four starts in two seasons with the AFC North franchise.