GREAT AMERICAN museums have, to put it bluntly, only two ways of acquiring major works of art: They can spend or they can stroke; they can buy or they can beg.

Consider, for example, the impressive exhibtion of 20th-century art opening today in the National Gallery's East Building. Its Picassos are terrific, its Picabia superb. Its Legers, Klees and Man Rays include first-rate examples. But one Morton G. Neumann is the true star of this show.

He manufactures cosmetics in Chicago. He is 82 years old. And he is -- how should one put it? -- a prime potential patron, one of the most important American collectors still on the loose today.

The man is being courted, there is no doubt of that. Collector Neumann is involved here in competitive flirtation, that most delicate of dances, the gallery gavotte. Its participants tred lightly, they smile and they bow.

The ritual, though far from new, has intensified of late. Once, when they were wealthy, haughty art museums purchased works they wanted; increasingly today they seek to buy with compliments, with catalogs and cunning, with high praise and with honors what they once bought with cash.

These matters are most sensitive. Gallery officials, while wishing it were otherwise, will hurry to inform you that no deals have been struck, no pledges have been made, that they hardly dare to hope that at least a few of collector Neumann's pictures, now on temporary loan here, will return to stay.

What makes this ritual so poignant is that it often fails. Rare the gallery director who has not been jilted by one collector or another. Two decades ago, Joseph H. Hirshhorn flirted very broadly with half a dozen cities before he settled upon this one. Calouste Sarkis Gulbenkian, the late Armenian oil man, broke as many hearts. Kenneth Clark, director of the National Gallery in London in the 1930s, once agreed to build a Gulbenkian museum on Trafalgar Square. In the '50s, when John Walker was director of the National Gallery here, he planned to do the same on what is now the site of the Gallery's East Building. Walker thought Gulbenkian's collection "the greatest in breadth and standard of quality assembled by one person in our time." "Museum directors," Walker wrote, "are predators by nature; no prey seemed as tempting as the Gulbenkian collection, no prize as desirable as some future Gulbenkian Foundation. I was determined to get both." He failed. The Gulbenkian Collection went eventually to Lisbon. Walker, in his book "Self-Portrait with Donors," speaks of Calouste Gulbenkian as "The Collector Who Got Away."

Neumann is a lesser catch, though his large collection of avant-garde art is one of the finest still in private hands.

He owns perhaps 600 works of modern art, of which 137 have been chosen for this show. The collection he began to form in 1948, when he sailed off to Paris to buy his first Picasso, surveys modern art from 1909 to 1980. And he is still buying.

Neumann's huge collection is particularly rich in works that were produced by artists he befriended. He knows Dubuffet and Miro, he gave watches to Picasso. "I had this idea," Neumann said recently. "P-a-b-l-o-p-i-c-a-s-s-o -- that's 12 letters, so I had wristwatches made up with the letters in place of numerals. I gave six of them to Picasso. I know that he liked them very much, because he saved them and only gave them to bullfighters he admired." One of his Man Rays, an object called "Self-Portrait," is inscribed on the back "For Morton Neumann with thanks for my automobile and the diamond ring for Julie." (Neumann had bought both in a New York dimestore. The "diamond" ring was plastic, the car a wind-up toy.

The Neumann show includes paintings by the cubists, the dadaists, the futurists, the pop artists, the photorealists, and the pattern painters, too. Will he put them on the market, leave them to his heirs, give them to museums? "I have no plans at present," Neumann said last week.

Neumann understands full well the dance that he is dancing -- and that underneath the niceties of any such endeavor is an implied quid pro quo.

He says it is an honor to have his objects shown at the National Gallery of Art. (Though his home-town museum, The Art Institute of Chicago, has been trying for some time to mount a Neumann show, they will have to wait until it closes here.) He knows that the attention given his exhibit will surely not diminish the value of its works. He says he is delighted that a fully illustrated, two-volume catalog of "The Morton G. Neumann Family Collection: Selected Works" accompanies the show. (Its tone is far from critical.Writing in the catalog, E. A. Carmean Jr., the curator in charge, feels it wise to tell the reader that as soon as he met Neumann, "I immediately realized . . . that I was going to like him very much.")

But what may the Gallery hope for in return?

Its East Building on the Mall was built to lure collections. To fill out its skimpy survey of 20th-century art, it sorely needs examples by the dadaists, the futurists, and the pop artists, too, for its holdings in these areas are embarassingly thin.

"There are things that Neumann owns for which we'd give our eye teeth," says an official there. "You will sense in this exhibit our recognition -- and our thirst." One curator involved says, "His collection would complement ours perfectly -- no one can fail to see that."

To read Carmean's description of his initial visit to the Neumann townhouse on Chicago's East Division Street is to feel the pang of curatorial yearning: "Straight ahead, eight Picassos spreading out across the wall. I can still recall the dizzying sensation as my eye beheld in a sweeping survey, a Klee and a Gris over my shoulder, the Picasso sculpture below on the table, then Picabia on another wall, as well as works by Mondrian, Delaunay, Balla, Ernst, Duchamp-Villon . . . and this was just the first room . . . The dining room. Seventeen Miros. All major. Nowhere else in America is there a room of this intensity. Next, up the stairs -- past Matisse -- to a study. Three important Legers, two Giacomettis, a huge Flack, 17 Klees . . . A room of surrealist art. . ."

Perhaps the best way to explore the Neumann exhibition, in Washington at least, is to do so with a want list.Imagine you could pick, say, a dozen objects for the Gallery's collection. Which ones would you choose?

The Gallery, it's true, owns lots of Picassos. But Neumann's are so fine one could not resist taking at least one -- say, the "Woman in an Armchair" of 1944. A semi-futurist Leger, "Woman Sewing" (1913), is comparably grand. Because the Gallery's collection is particularly weak in objects by the dadaists, Francis Picabia's "Amorous Parade" (1917), must rank among the most important objects in this show. The Man Rays here are made of metronomes and shoe trees, masks and coiled springs. Perhaps, if one could bend the rules, and count that display case of Man Rays as a single work, one could have them all. Paul Klee's large and late "Capriccio in February" (1938) -- which Neumann lent the Gallery when the East Building opened -- is another must. It would be far from easy to choose from these Miros. Should one take one, or two, or three, or, instead, the whole roomful? Don't be greedy, go for one. Pick the "Spanish Dancer" of 1929, an assemblage that includes cork, sandpaper and string. And take that bronze and skinny Giacometti dog. The Dubuffets, as well, are moderately tempting; pick one of the portraits. The early Guston, too, is fine, though he can keep the Kline.

Forget the Neumann minimalists, his Carl Andre floor piece and his box by Donald Judd, and do not take his Noland or his so-so Morris Louis. But do not let his best pop art escape. Take his early Robert Rauschenberg, "Factum II" (1957), and his case of Oldenbergs, his Roy Lichtenstein -- and his Andy Warhol, a work of 1963 called "The Men in Her Life." (The "her" is Elizabeth Taylor. Warhol based his painting on a photograph that shows her with Mike Todd and Eddie Fisher. There are, Carmean points out, five now-famous figures portrayed in this picture: Liz, Mike, Eddie, Debbie and Princess Leia, still in swaddling clothes.)

The photo realists are perhaps too well represented in Morton Neumann's show. His Audrey Flack, for instance, is a truly vulgar picture, and his huge Chuck Close would be easy to pass by. But that early Richard Estes, with its overtones of Hopper, and that batch of cabbages painted by Ben Schonzeit would be nice things to have.

Nobody but Neumann is able now to say whether all -- or any -- of these works will eventually belong to the National Gallery of Art. But there is no doubt that the gift of two or three would be enough to justify the mounting of this show. It closes New Year's Day.