The sellers, who have owned it since 1932, are asking for four million euros, and the Louvre has raised three-quarters of the amount.
"It needs one last effort for this painting to join the national collections," Louvre president Henri Loyrette says on the French-language website.
The small-sized work, painted in 1531 and always privately owned, shows three women against a dark background, wearing nothing but necklaces, apart from the central figure who also sports a red hat.
According to the website, the painting is remarkable for its "astonishing perfection", its great rarity and its state of conservation, meriting its designation as a national treasure.
Donors have until the end of January to contribute whatever sum they wish to the purchase of the painting, and see their names figure on a list of sponsors.