The former Boro and England defender has been among the bookmakers' favourites to succeed Tony Pulis, under whom he worked as first-team coach last season, since the Welshman's departure at the end of the campaign.
Both the Press Association and Sky Sports are reporting that chairman Steve Gibson is ready to appoint the 39-year-old, and the club is currently engaged in the process of putting together his coaching staff.
Woodgate made a total of 112 appearances for his home-town club in two spells and ended his playing career there in 2016.
He returned as a coach under interim boss Steve Agnew in March 2017 and worked with the Under-18s before being elevated to the senior set-up once again by Pulis.
Meanwhile, Woodgate's former Leeds and Tottenham team-mate Robbie Keane confirmed to reporters in Ireland on Wednesday that he had spoken to Boro about the assistant manager's role, although it is understood he is only one of a series of candidates to be sounded out.
Keane, who almost signed for the club during Bryan Robson's reign as manager, is currently working under Republic of Ireland boss Mick McCarthy as one of two assistants along with the manager's long-time ally Terry O'Connor, and would combine the two roles.
He scored 68 goals in 146 appearances for his country - both Irish records - before retiring from international football towards the end of a career which also included spells with Wolves, Coventry, Inter Milan, Leeds, Tottenham, Liverpool, Celtic, West Ham, LA Galaxy, Aston Villa and ATK Kolkata.
"They want me to come in as assistant manager and it's up to me to think about it," Keane told reporters in Crumlin on Wednesday. "It won't affect me here, which is key.
"I spoke to Mick McCarthy about it and said under no circumstances would I take it if I had to leave Ireland, and he was delighted with that.
"I've known Woody for 20 years and he's a good friend, but I have to think about it."
In addition, it is understood Leo Percovich, who worked as goalkeeping coach at the Riverside Stadium under Aitor Karanka, has been sounded out to return to a similar post.
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