The two full-length portraits of a young couple painted shortly before their wedding in 1634 have been owned by the wealthy banking family since the mid-19th century and have rarely been seen in public.
Culture Minister Fleur Pellerin said the offer would be "exceptionally financed by the Bank of France".
Pellerin and her Dutch counterpart Jet Bussemaker wrote to the Rothschilds in July proposing the Louvre and Rijksmuseum in Amsterdam each buy one painting.
On Monday, the Netherlands said it was hoping to buy both artworks, with the Government footing half the 160 million bill, while the Rijksmuseum would raise the rest.
However, the joint acquisition seemed back on the table in what Pellerin described as "an innovative solution that would strengthen the cultural co-operation between France and the Netherlands".
She said the two artworks would alternate between the museums.
The artnet website said the portraits have been publicly viewed once in the past 150 years.