Nafea Faa Ipoipo, or When Will You Marry?, was owned by a Swiss collector.
The oil-on-canvas was produced in 1892 during Gauguin's first visit to French Polynesia. It features a pair of Tahitian girls seated next to a tree.
Unconfirmed reports suggest it was sold to a museum in Qatar for nearly 300 million (197 million pounds).
The small oil-rich state paid the previous highest price for a painting, a work by Paul Cezanne which sold for a reported 158 million pounds.
For decades it had been on loan to the Kunstmuseum Basel but Staechelin decided to sell the painting after a disagreement with the museum, US media report.
Staechelin, a retired Sotheby's executive, told the New York Times he would not divulge the identity of the buyer.
It was not immediately clear where the sale had taken place.
However the paper, which first reported the sale, quoted sources saying the painting had been sold to Qatari buyers.
Officials in Qatar have not yet confirmed the purchase.
Sheikh Saud bin Mohammed Al-Thani, a former minister of culture who died last year, lavished more than 1 billion dollars of the country's money on artworks, the report said.
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