The 25-year-old only moved to Upton Park from Sochaux in the summer, signing a four-year contract with the newly promoted side.
He opened his Hammers account in a Capital One Cup tie against Crewe in August and also scored a memorable goal against Chelsea in October.
However, he has struggled to command a regular first-team place, making just two starts so far this campaign.
Bordeaux coach Francis Gillot worked with Maiga at Sochaux and was believed to be targeting him in the January transfer window.
West Ham boss Sam Allardyce was willing to let the striker make a temporary move away from the club this month.
However, the Hammers were unwilling to pay part of his wages while he was away from east London, causing the deal to collapse.
Bordeaux chairman Jean-Louis Triaud said: "Maiga won't come. West Ham didn't want to help for a loan. So, Maiga won't become a Girondins player."