The 24-year-old failed to make a single Premier League appearance for the Gunners after joining in 2010.
Work permit issues saw him loaned to a series of clubs including Levante, Alcoyano, Ponferradina, Murcia and Almeria.
Arsenal went on to offload Wellington to Fluminense last summer but included a buy-back clause of £3m in his contract, which has now been triggered.
The French side are reportedly paying £4.3m for the Brazilian winger and had an agreement in place with the North Londoners last week.
Bordeaux manager Jocelyn Gourvennec is quoted by The Express as saying: "Wellington Silva is expected to arrive this weekend and sign with us early next week after his medical visit.
"It took a bit of time, but he should be with us. After that we will still have some work to do but I will not detail this now."
Last term, Wellington made seven appearances for Fluminense, scoring one goal and providing one assist in all competitions.