As soon as Richard Grant, executive director of the Diebenkorn Foundation, glimpsed the three drawings in an Upper East Side apartment several years ago, he knew there was a problem.

The artwork on the wall had been previously identified by the foundation as fake Richard Diebenkorns. But here they were again, proudly displayed as Diebenkorns by a new owner who had no idea he had bought discredited drawings.

For organizations like Grant’s that are charged with protecting an artist’s legacy, the job of patrolling for fakes has become something like a game of Whac-A-Mole.

“You put it down, and then five, seven years later, poof!, and there it is again,” he said by phone from the foundation’s offices in California.

The resale of fakes is a persistent problem without a good solution, say collectors, dealers, artist estates and law enforcement agencies. Although the Federal Bureau of Investigation can seize forgeries in criminal cases, these represent only a tiny portion of the counterfeit art that is circulating.

“They churn through the market,” James Wynne, an FBI special agent who handles art forgery cases, said of fakes.

There are no clear rules for what happens to phony art after it is identified. “It all depends what the facts are, what the art is, how many works are involved and how expensive they are,” he said.

Art whose authenticity is disputed occupies a special sort of limbo, as demonstrated by the recent settlement between Knoedler & Co., a Manhattan gallery that abruptly closed in 2011, and a customer who accused that gallery of selling him a forged Jackson Pollock for $17 million.

The FBI is investigating whether that painting, known as “Silver Pollock,” might be part of a larger cache of forgeries. But no charges have been brought and the gallery maintains that the work is authentic. So what happens to a $17 million painting that some people consider a fake?

Given the publicity surrounding this particular case, a sale any time soon would be surprising, art lawyers and dealers agree. But nothing in criminal law would necessarily prevent the owner from selling it today as a Pollock. (Details of the settlement are confidential, including who owns the artwork now.)

When it comes to undisputed fakes, law enforcement officials try to halt resales by stamping such works as fake or, in rare cases, destroying them. Each option has drawbacks, including the possibility of mistakenly destroying an authentic work.

Jack Cowart, executive director of the Lichtenstein Foundation, said that during the years that the foundation authenticated works by Roy Lichtenstein, he regularly noticed that collectors informed that they had a fake would later quietly sell it as genuine. “And then we’d find somebody else would send the same work to us six months later” asking for it to be authenticated, he said.

In France, Switzerland and other countries that recognize the “moral rights” of an artist, heirs or foundations like Lichtenstein’s can ask the courts for permission to destroy a fake. But Ronald D. Spencer, a Manhattan lawyer and editor of the art-law handbook “The Expert Versus the Object: Judging Fakes and False Attributions in the Visual Arts,” said he was glad that is not done in the United States. The notion is anathema, he said, noting how frequently opinions about authenticity can change.

Just a few months ago, three J.M.W. Turner paintings that had been dismissed as fakes were reclassified as genuine. “Stamp it, by all means, so that any subsequent owner knows that it was considered a fake, but don’t destroy works,” Spencer said, echoing the view of many art dealers. If destruction becomes routine, he added, a genuine work will mistakenly be consigned to the shredder at some point.

Marking a work was the course taken in 2011 by the Dedalus Foundation, a nonprofit created by the artist Robert Motherwell, after it identified a putative Motherwell painting as forged.

As part of a civil settlement, Dedalus demanded that the work be permanently marked as a fake. Now on the back of the work, “Spanish Elegy,” an indelible stamp states that “this painting is not an authentic work by Robert Motherwell but a forgery.”

Once law enforcement becomes involved, the picture changes. Government agencies like the FBI, the U.S. attorney’s office or the Postal Service (which has a role in forgery cases involving mail fraud) may stamp forgeries after a conviction or plea agreement, federal prosecutors in Manhattan said.

The government has the power to destroy uncontested fakes or to ask a judge for permission to do so if the defendants insist they are real, but that rarely happens, law enforcement officials said.

Judges, however, can be unpredictable.

In 1995, when a gallery in Hawaii could not pay a $2 million fine levied for selling phony prints by Salvador Dali and others, a federal court ordered the government to auction off more than 12,000 of them to the public.

The proceeds went toward repaying the debt, but federal investigators and art dealers protested the large-scale sell-off of fakes. Most of the prints were marked with only a tiny stamp identifying them as counterfeit, which was easily hidden by a frame. Others had a removable sticker or no marking at all to identify them as discredited.

“This makes the federal government an accessory to future art fraud,” Bernard Ewell, a Dali art appraiser, said at the time. Then he added, “But I’m delighted because it gives me guaranteed job security.”

As for the phony Diebenkorn drawings that Grant saw in that Upper East Side living room several years ago, their whereabouts today are a mystery. The owner told him that he had returned the forgeries to the dealer and gotten his money back.

DANA OLAND’S column will be back next week.