The National Gallery of Art is beginning to prepare for life after the Mellons.

For the first time in its history, the gallery is asking rich Americans--other than the Mellons--for large amounts of cash.

Individual donations--of at least $1 million each--are now being solicited. The aim is to assemble a "Patrons' Permanent Fund" of $50 million--whose principal will not be touched, whose interest will be used for buying works of art.

J. Carter Brown, the gallery's director, says that $15.6 million "in money and hard pledges" has been raised since the three-year fund drive, never publicly announced, began late last summer. "That is more," he added, "than many other institutions ever hope to see."

"We contact people of wealth," said John C. Whitehead, a member of the fund-raising committee. "Of course we mention Mellon. That's part of the pitch. The National Gallery is a great national institution that now, for the first time, needs broad national support."

Still, even $15.6 million seems a small amount when it is compared with the huge sums the Mellons have provided in the past. Many other donors have given works of art, but founder Andrew Mellon, and his children Paul and Ailsa, have always been the gallery's main source of private cash. Andrew Mellon built the gallery's West Building, for $15 million, in 1941; an additional $5 million, given by Paul Mellon, is now being spent on rebuilding its ground floor. The $95-million East Building, which opened to the public in 1978, was built with Mellon monies. And the most costly pictures the gallery has purchased--its $2-million Jackson Pollock, its $5-million Leonardo, and a dozen more--were bought with funds provided by the late Ailsa Mellon Bruce.

Paul Mellon, now the gallery's board chairman, made the initial gift to the new $50-million Patrons' Permanent Fund--a donation of $5 million. Does the gallery's decision to turn to other donors mean that Mellon is weaning the museum? Brown, when asked if he expects Paul Mellon to give more, made a sudden, slicing motion with his hand. "From all one can gather," Brown said, "he very much wants the gallery to fly on its own wings."

"I wouldn't say Paul Mellon is weaning the gallery," said gallery president John R. Stevenson. "But he knows it needs additional support, and that his sister's gone. Inevitably, he is winding down."

Reached yesterday in Manhattan, Mellon said, "absolutely not" when asked if he is "winding down" his long commitment to the gallery. "My commitment isn't winding down. I am," said Paul Mellon. "After all, I'm going to be 75 in June.

"I have made a substantial contribution to the fund . . . If the place is to remain alive its director and its curators must feel they have some input in what the gallery acquires . With the increasing costs of works of art they can no longer rely on getting a few dollars here and a few dollars there. I have given them pictures. And I intend to give them more."

The gallery has never used its federal appropriations for buying works of art. Though rich in pictures and in buildings, its purchase funds, says Brown, "are now pretty flat." Though he says there is "about $4 million left" in the Ailsa Mellon Bruce Fund, that may not be enough to buy a single major painting, given today's prices. The gallery does have other minor purchase funds, some of them restricted, with a total value of perhaps $3 million, but it feels it needs far more. Hence the new campaign.

Its eventual success in many ways depends on perceptions of Paul Mellon. Some prospective donors may see no need to give, suspecting that the gallery can always turn to Mellon. Others may well feel, as Stevenson observes, "it is time we did something for Paul. He has already done so much. And he is, after all, not the only rich person in the country."

Mellon, as board chairman, is actively involved in the gallery's endowment drive. A seven-member campaign committee has joined the trustees in seeking large donations. Its members include Carter Brown; R.L. Ireland III of Brown Brothers Harriman & Co., the New York investment banking firm; Bruce B. Dayton, former chairman of Dayton-Hudson, the Minnesota merchandising company; Robert H. Smith, president of the Charles E. Smith Building Corp., the developers of Crystal City; Miami lawyer Louis J. Hector; John C. Whitehead, senior partner in Goldman Sachs & Co., the Manhattan investment banking firm; and Katharine Graham, chairman of the board of The Washington Post Co.

Mellon has in recent years made other large commitments to American museums. His own remarkable collection of English art is housed in a museum, designed by Louis Kahn, which Mellon built for Yale. And he yesterday acknowledged that he will give $6 million, as well as many pictures, for a new wing to be added to the Virginia Museum in Richmond.

Last week the Virginia legislature voted $10 million to complete the funding package for the new addition. The wing, to be designed by Hardy, Holzman, Pfeiffer, will cost $22 million. Mellon said yesterday that Virginia art collectors Frances and Sydney Lewis will also give $6 million. Mellon also said that he intends to give to that museum some English sporting pictures and a number of French paintings, among them the 59 from his collection--by Degas, Cezanne, Gauguin, Corot and Picasso and other well-known masters--that are now there on loan.

The fund-raising committee for the National Gallery's endowment has begun making presentations in half a dozen cities. "We are only beginning to make presentations, and because we're asking for large amounts of money--large amounts even for wealthy people--we've had few decisions, either positive or negative . . . Most of those we see already have some close association with another art museum--but that's part of our pitch," said Whitehead. "We tell them the National Gallery is different. It's the National Gallery. There are certain pictures that should belong to the nation, and those are the pictures they should help to buy."

"I don't ask strangers for contributions," said committee member Smith. "I go to people that I know."