At the time, I was chairman of the classics department at the University of California, Los Angeles, which had been invited by the Italian government to reopen the excavation of the Villa of the Papyri in Herculaneum, buried by the eruption of Mount Vesuvius in 79 A.D. The site hadn’t been explored since the mid-18th century, when a partial excavation yielded many important works of art and more than 1,800 papyrus texts, and there was undoubtedly more to be found. The only problem was that the estimated budget was $20 million, far beyond what my department could afford.

Right away I thought of the Getty. It had the money and, I hoped, the interest: after all, one of the first museums J. Paul Getty built, called the Getty Villa, is a copy of that same Herculanean villa. So I met with Ms. True, then a junior curator, to suggest that the Getty sponsor the dig. She quickly rejected my idea  it would violate museum policy, she said, to have anything to do with an excavation. If anything were stolen from the site, the museum would be the natural target of the authorities’ suspicions. Some years later, the Italian government moved forward with another partial excavation of the villa.

In all the years since, to my knowledge, no American museums have tried that kind of partnership. But it made sense then, and it still makes sense now. Excavation is the lifeblood of archaeology. Without it, museums can only recycle exhibitions of well-known masterpieces. And despite two centuries of digging, much more remains to be discovered than has yet been found.

If only ownership could be separated from possession, then museums might strike a deal with countries like Greece and Italy. Here’s how it would work: The countries of origin would own anything that was excavated there and keep most of the finds on display in local partnering museums. But the museum that sponsored the dig would be allowed to borrow a percentage of the finds and exhibit them in America. Eventually, all the finds from a site would be exchanged on a rotating basis between the country of origin and the museum, which would pay the expenses and insurance.

Even individual collectors could invest and participate in the exchanges, if they were trained to care for the finds on temporary loan to them. Someday, investors or their heirs could sell these shares at auctions and galleries, just like works of art. In this way, all the stakeholders in today’s antiquities market could be part of the new deal.

Where should museums and investors begin? Well, there’s the tomb of Antiochus of Commagene at Mount Nemrut in southeast Turkey. The location of the giant burial mound is well known, but no one has had the money to find the king’s tomb within it. A deal with the Turkish government to finally excavate the chamber could yield invaluable information on the interactions of Persian, Greek and Roman cultures in the second and first centuries B.C. Then there’s the Temple of the Divine Augustus in Rome. Here, too, the site’s likely location (under a street next to the Forum) is known, and ancient sources suggest that the monument was spectacular in design and décor.

Farther afield one could add Pataliputra in India, reputed to be the most populous city in the world in the third century B.C.; or the tomb of Qin Shi Huang, China’s first emperor, who died in 210 B.C. Qin erected thousands of realistic terra-cotta statues of soldiers around his tomb; these Xi’an Warriors have been famous since they were first discovered in 1974. But the excavations have thus far touched only a small, peripheral part of the site. The emperor’s enormous mausoleum has been located but not yet explored. We probably wouldn’t find everything that ancient sources say is buried inside  cross-bow booby traps, model palaces and 100 small rivers flowing with mercury. But we would find plenty. Even the famous Pompeii remains a mystery, with a third of the city still underground.

Finds from these sites and the scores more like them around the world have filled many rooms in our museums and have contributed enormously to our understanding of everyday life in antiquity, yet we have much more to learn. It’s been 25 years since my conversation with Marion True, but Pompeii and Pataliputra have been waiting far longer. It’s not too late for museums to start digging.