In a club statement,, external Stockport said Mangan and Davide Ancelotti had previously worked together when completing their coaching badges.
Last September, Mangan was denied a move to Real Madrid to work with Carlo Ancelotti after he had a work permit rejected by the Spanish government.
Botafogo have recently returned home after reaching the knockout stages of the inaugural Club World Cup, where they were beaten by fellow Brazilian club Palmeiras in the Round of 16 in extra-time.
Despite beating Paris Saint Germain on the way to the last 16, Botafogo sacked head coach Renato Paiva last week with the Portuguese claiming that owner John Textor, who also co-owns Crystal Palace and French club Lyon, made attempts to pick the team.