The former Celtic boss (52) succeeds Gareth Southgate, who was sacked last week when Gibson's two-week chase for Stachan ended on Tuesday.
Boro beat Derby 2-0, but it was not enough to save Southgate's job.
Boro website: "Gordon Strachan will be unveiled as Boro's new manager at a press conference at the Riverside at 2.30pm this afternoon."
Strachan began his managerial career at Coventry before moving on to Southampton, who he led to the FA Cup final in 2003.
He took charge at Celtic in May 2005 and led the team to three successive Scottish Premier League titles.
Gibson: "When I sat down and considered our start to the season, I felt that the league table was actually more favourable than some of our performances.
"Then I looked at the important games against the teams immediately around us, and took those results into account.
"In general the results and the performances have not been to the level I would have hoped for."