Bordeaux, who won the last of their six first-division titles in 2009, were sanctioned on July 9 by French football's financial watchdog, the DNCG, as they needed to find 40 million euros to balance their books.
They had been in talks with the owners of Liverpool, Fenway Sports Group (FSG), before the American investors pulled out of negotiations earlier this month.
"Even if discussions had restarted in recent days, representatives of FSG told Bordeaux and their majority shareholder yesterday, Monday July 22, it no longer wanted to pursue talks despite guarantees brought by the different interested parties," Les Girondins said in a statement.
"Due to the lack of new information, Bordeaux have withdrawn their appeal against the DNCG's decision of July 9, 2024.
"In result, Bordeaux accept the punishment of an administrative relegation to National 1 for the 2024/2025 season and will once again have to present its financial budget to the DNCG.
"The club and its shareholders assure supporters, and all interested parties, their desire to protect the club's interests," they added.
Bordeaux, whose former players included Zinedine Zidane and Bixente Lizarazu, were relegated to Ligue 2 in 2022 after finishing 20th in the top flight.
They have not played in the third division since becoming professional in the 1930s.
The new National season begins on August 16 with Bordeaux expected to play at their 42,000-capacity Matmut Atlantique home, France's sixth-biggest stadium, for the campaign.