The former Real Madrid defender is the favourite to be named as Tony Pulis' successor, according to BBC Tees Sport.
Woodgate, who was part of the coaching set-up under Pulis, has never been a manager of a club before, but will reportedly take over the reins at Riverside Stadium ahead of the 2019/20 season.
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Furthermore, Woodgate's former Leeds teammate, Robbie Keane, has reportedly also been in talks with the club to work alongside the 39-year-old at Boro as they set their sights on a return to the English Premier League.
Pulis parted company with Boro at the end of last season after the club failed to reach the playoffs, finishing seventh on the Championship log standings.
Woodgate made 112 appearances for Boro as a player.