One of the few paintings by the English painter J.M.W. Turner to remain in private hands failed to find a buyer at auction here today, even though it initially appeared that an unidentified woman had purchased it for $1.815 million.

"Landscape With Walton Bridges" (1840-1845), which was consigned for auction from the estate of Henry S. Morgan, grandson of the financier J. Pierpont Morgan, was featured here at Sotheby's in a separate sale at the end of a morning sale of 19th-century paintings that was otherwise uneventful.

John Marion, today's auctioneer and president of the New York branch of the London-based art auction house, announced the abortive sale in a statement issued three hours after the hammer fell: ". . . The painting just missed selling. There have been a number of inquiries after the sale. Negotiations are going on now, and we think it will be sold later." The painting, Sotheby's later said, failed to meet the reserve (minimum) price the consigner had set.

In May 1980, Turner's "Juliet and Her Nurse" (1836) established the auction record for any work of art when it brought $6.4 million. Last July, "Temple of Jupiter--Panellenius Restored" (1816) was sold at Christie's in London for $1.1 million, and many experts regard that as a strong price for Turner in today's market.

More than 500 people attended the sale, although many of the seats remained empty. Members of the Morgan family were reported in the room.

In less than four minutes, the painting had reached what later turned out to be a stalemate at $1.815 million (which includes the 10 percent buyer's premium). A quick round of applause followed, and the bidders, including the unidentified woman who placed the final bid, left quickly.

One New York art dealer, Richard Feigen, who is a specialist in Turner, said after the sale: "There were some problems with the picture. It needed restoration. And it was not one of Turner's great epic disaster scenes -- fires, avalanches or floods which 'Juliet and Her Nurse' was."

Said David Nash, head of the painting department at Sotheby's: "I guess we didn't have auction fever in the room today."