The deal will see owner Gerard Lopez exit the club after a five-year spell that has seen the 2008-09 Ligue 1 winners descend from the top tier of French football to the sixth, where they will now attempt to start their season.
"This operation (sale) is concluded for a symbolic euro with the assumption of debts," said a statement from Bordeaux, which called it "a major step for the future" of the club.
The debts Park Bench will assume are close to €18 million, said French media outlet Ici.
Park Bench is a data analytics and software company with an investment portfolio. It is owned by American Evan Sofer and Briton James Bord.
The fund already holds shares in Spanish second-division outfit Cordoba and is the majority shareholder in Scottish Championship (second-tier) club Dunfermline Athletic.
Bordeaux have twice been placed into administration since 2021 and as a result of relegations enforced by the National Directorate of Management Control (DNCG), have spent the last two seasons playing in the Championnat National 1 (fourth tier).
This year, Bordeaux have faced more financial issues, which led the DNCG - which oversees the accounts of professional clubs in France - to relegate them to Regional 1 (sixth tier).
The Regional Management Control Commission of the Nouvelle-Aquitaine Football League must now approve the sale and the club's budget for 2026-27 for Bordeaux to play in Regional 1.
The Bordeaux Commercial Court must also approve the plan so Bordeaux can avoid liquidation.