The Portrait of Dr Gachet by Vincent van Gogh was bought by Ryoei Saito, a businessman, at Christie's in New York in 1990 for $82.5m (pounds 55m), the most paid for a painting.
Mr Saito once told friends that the painting should be cremated with him when he died so his children would avoid colossal death duties. He died three years ago.
The missing work is one of two portraits by Van Gogh painted of Paul- Ferdinand Gachet, a doctor who treated him at the end of his life. It has long been considered the last important piece painted by the artist before his death in 1890.
The alarm was sounded by the Metropolitan Museum of Art in New York, which had hoped to include the work in a current exhibition of Van Gogh portraits. But its curators were unable to locate it. It was then that stories of Mr Saito's unorthodox cremation request came to light.
Mr Saito, the former chairman of the Daishowa Paper Company in Tokyo, apparently suggested that the Van Gogh and a Renoir, purchased at the same sale, should be burnt with his body.
Nobody has given up hope that the work may simply have changed hands and that the new owner has chosen to remain anonymous. There is no indication, however, that it has been seen since Mr Saito's death from a stroke in 1996. He apparently made the cremation remark to friends after being hit in 1990 with a tax bill of $24m.
The exhibition catalogue at the Metropolitan Museum of Art explains the absence of the portrait with the note: "Present location unknown." A spokesman said: "We tried to borrow it for our Dr Gachet show, but we couldn't find the owner. We don't even know who sold it." It is not even clear that Mr Saito much enjoyed the Van Gogh when he was still alive. It is reported that he looked at the picture and the Renoir, for which he paid $78.1m, just once before ordering that they were stashed in a warehouse.
Previous owners of the work included Hermann Goering, who acquired it for Hitler's collections.
A spokesman for the Daishowa Paper Company admitted that, as the painting belonged to Mr Saito and not to the company, staff were unsure where it was now.
However, they had heard that it had been sold on to somebody else. The spokesman added: "There's no fact to the rumour that he brought the painting to his coffin. Somebody owns that painting."