Cupid Complaining to Venus, by Lucas Cranach the Elder, the German artist, was bought by the gallery in 1963 and is worth millions of pounds today.
If claimants to the work, such as descendants of the original owners, come forward, it will go into an official process to determine if they are entitled to have the painting returned or to receive a payout.
The National made its admission after it was told that the painting had been taken from a warehouse in southern Germany in 1945 by Patricia Lochridge Hartwell, an American reporter. The information came to light when Hartwell’s son Jay contacted the gallery.
The painting was almost certainly stolen by the Nazis from Jews and may have been one of thousands of works looted by Hermann Goering, Hitler’s deputy.
“We are acknowledging that there might be a claim on this painting,” said Charles Saumarez Smith, director of the National. “We simply felt under an obligation to make it public that there is a question mark.”
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So far six works in British galleries — including the Tate and the British Museum — have been identified as Nazi loot and resulted in payouts. The Cranach would be by far the most valuable, although the claimants would not receive its full market value under restitution rules.
Cupid Complaining to Venus, which depicts the young god telling the goddess that he has been stung by some bees, was painted in about 1525 by Cranach, a friend of Martin Luther, the theologian who inspired the Reformation.
Although it is not certain that the work was owned by a Jewish family, there are no official records of its whereabouts from 1909 to 1945.
After the Nazis came to power in Germany in 1933 they began stealing the possessions of Jews. The process accelerated in the war, when museums across German-occupied Europe were also plundered.
Goering was the most avid looter and appointed agents throughout Europe to scour homes and galleries for plum items to add to his collection, including paintings, sculptures, Roman artefacts, jewellery and tapestries.
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If the Cranach, one of nine by the artist owned by the National, had been taken from a museum rather than a private individual, it would have been catalogued and is likely to have been returned after the war.
According to the next issue of The Art Newspaper, Hartwell was reporting in southern Germany at the end of the war.
It is possible that the Cranach was taken to the store from Schloss Neuschwanstein, a 19th-century castle built in the Alps by King Ludwig II of Bavaria. Another possibility is that it came from a Goering cache assembled after the war at Unterstein, near Hitler’s mountain home at Berchtesgaden.
A press conference was held there in mid-May 1945 to publicise the collection and it is thought that Hartwell, who had trained before the war at Columbia University’s school of journalism in New York, attended.
Cranach’s work was highly regarded by the Nazis, with Goering being a particular fan. Another link with Goering is that Hartwell is believed to have visited one of his homes in May 1945 and removed one of his military sashes. When she took it back to America she turned it into a hat and a handbag.
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Hartwell is thought to have taken the work when she was allowed into a storehouse full of art by the American soldiers guarding it and was invited to take her pick.
After the reporter returned to America with her painting, she began working in New York as a press officer for Unicef. In 1961 she and her second husband approached the Metropolitan Museum of Art, offering to sell the Cranach, but the sale did not proceed.
It was then passed to the New York dealer E & A Silbermann, which in 1963 sold it to the National Gallery. There are no papers from Silbermann as it has long been out of business and no records of sales were passed on.
According to the National, Silbermann said that it, as a dealer, had purchased the work in 1909 from ancestors of the Hartwell family.
However, it is believed that the painting was sold in 1909 to a different buyer by its then owner, Emil Goldschmidt, a collector from Frankfurt, through Rudolph Lepke, the Berlin-based auctioneers.
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Two years ago the National Gallery was e-mailed by Jay Hartwell, who works at Hawaii University, seeking information about the painting.
His mother, who moved to Hawaii where she became director of the state’s arts council, had died in 1998.
Jay Hartwell, who met officials from the gallery last year, told the National some of the background to the story which he had picked up from his mother.
There is no suggestion that the Hartwells acted in bad faith in 1945 or the National in 1963 when it purchased the Cranach. But in recent years there have been growing efforts to restore to Jewish families the possessions which had been looted by the Nazis.
Speaking this weekend from his home in Hawaii, Hartwell said his family “wanted to try to find out more about the painting and its background. My mother was very interested in art but for personal reasons at this time I cannot comment any further”.
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Any claimant’s case would be considered by the Spoliation Advisory Panel, set up in 2000. It can order financial compensation, usually a compromise figure between a work’s worth at the time of its loss and current value, or the return of the work.