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Auction house Sotheby's has now confirmed that the overseas buyer has withdrawn an application to take the picture out of the country.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.

The proposal formed part of a public consultation by Dorset Clinical Commissioning Group (CCG) into a shake-up of health services in the county.
The plans have been criticised by opponents as "perverse logic".
CCG chief officer Tim Goodson said all the responses to the consultation would be considered.
Read more on this and other stories from across the south of England
The consultation took into account 18,500 questionnaires, 2,000 people at drop-in sessions, 1,000 people interviewed over the phone and some paid focus groups.
The most contentious element - to create a single special care baby unit (SCBU) covering the whole county for premature babies - was rejected by two-thirds of people.
More than 25,000 people have already signed petitions against the closure of Kingfisher children's ward at Dorset County Hospital in Dorchester.
Another proposal to offer planned care from a single hospital and emergency care at another was supported by half the respondents but people were split over whether the A&E should be at Bournemouth or at Poole.
Councillor Alan Gordon of Crossways parish, near Dorchester, said: "This is part of a long succession of financial cuts to the health service and it's about time these cuts were reversed. This is perverse logic and what we should be thinking about is funding the health service properly."
CCG chief officer Tim Goodson said: "We want to consider all the responses we've had and the strength of feeling by members of the public, not just on the paediatrics and SCBU issue but some of the community issues and around the acute care stuff."
A final decision on the proposals will be made later this year.
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.
[NEXT_CONCEPT]
Plans to cut children's hospital services in Dorchester and create a premature baby unit in east Dorset have been met with strong opposition.