Portrait of Caterina, dated to 1657, has been at Penrhyn Castle, Llandegai, Gwynedd, since 1860 before being provisionally sold.
But the UK government withheld the export licence until February.
This was to enable a UK buyer to raise the money to buy the painting.
Auction house Sotheby's has now confirmed that the overseas buyer has withdrawn an application to take the picture out of the country. It may be loaned to a museum or art gallery.
The subject of the portrait, Catrina Hooghsaet, was a wealthy Amsterdam Christian Mennonite who was married but separated from her husband.
In a statement, the Art Fund said the painting was of "supreme national importance" and its future was still "perilously unsafe".
It had started fund raising to buy the picture for the nation.
Art experts say it is one of the Dutch master's finest portraits.
A £35m painting by Rembrandt, which has been in Wales for 150 years, will stay in Britain for the time being after an overseas buyer withdrew an application for an export licence.