The 24-year-old failed to make a single Premier League appearance for Arsenal after joining in 2010, with work permit issues seeing him loaned to clubs including Levante, Alcoyano, Ponferradina, Murcia and Almeria.
Wellington returned to Fluminense in July 2016 but he has now been given another shot at European football after a deal was agreed with Bordeaux.
Bordeaux also joked the club is in talks with Wellington's fellow Brazilian Neymar, who is rumoured to be joining Ligue 1 rivals Paris Saint-Germain for a world-record fee of €222million from Barcelona.
Pour ceux qui demandent Neymar, les négociations continuent. Dès qu'il aura appris à dire "chocolatine" il signera. Pas avant.— FCGirondins Bordeaux (@girondins) July 23, 2017
"For those who ask about Neymar, the negotiations are continuing," Bordeaux posted on Twitter. "As soon as he learns to say 'chocolatine' he will sign. Not before."