The story is a classic of its kind, pitting two brash Americans against the custodians of the 250-year-old Sotheby Parke Bernet, the starchy, British-based fine arts auction house. "A national institution," the London Times recently called it, whose "survival in its present form matters to Britain."

Stephen Swid and Marshall Cogan are New York businessmen in their forties who have made it big in felt carpet underlay and designer furniture. Now they want to buy Sotheby's and if they succeed, says Graham Llewellyn, its present chief executive, "I shall blow my brains out."

What has driven Llewellyn and the rest of Sotheby's top management to such emotion in fighting off Swid and Cogan's takeover bid is the absence, they say, of "synergy" between themselves and the Americans--which, according to the Oxford English Dictionary, means compatibility of nerve centers or mental faculties.

The real problem, in Swid and Cogan's view, is something else--British snobbery. "We may not be as well-bred as they are," says Swid of his detractors, "We may not be from the landed gentry, but we are not the Philistines that Llewellyn has made us out to be." What he and his partner don't know about the auctioneering world, he insists, they can learn from experts, and while they do, Sotheby's will prosper anew from their hardheaded management.

Swid and Cogan, Ohio State and Harvard graduates respectively, are two plainly ambitious, hard-driving businessmen, who, having succeeded in the less-than-glamorous trade of making felt to put under carpets, started to branch out. In 1977, they acquired Knoll International, a respected furniture manufacturer. They joined committees at places like the Museum of Modern Art and the Guggenheim Museum and acquired art of their own to hang in their Park Avenue apartments.

"They lack only one thing for making it big in New York," Forbes, the American business magazine, noted in a profile. "Few people have heard of them." Lest British readers miss the point, the Guardian observed in an account of the battle for Sotheby's, " Swid and Cogan are still considered . . . to be on the fringe of the established banking and financial fraternity although well connected in a small band of entrepreneurs and particularly among Jewish circles."

It was last fall when, to the immediate chagrin of Sotheby's present board of directors, Swid and Cogan started buying shares in the art auction house. At the time, Sotheby's was in the midst of sharply scaling back its ambitious international expansion program, even closing one of its New York showrooms, because of money problems. Before long, Swid and Cogan had about 14 percent of the shares, the largest single bloc, and asked for a meeting with the senior management in London. It was, from all accounts, disastrous.

"Swid and Cogan have nothing to offer us," Llewellyn said then and has repeated many times since. "They know nothing about the fine arts auction world."

According to Swid, he and Cogan were told their interest and investment was the worst thing that had ever happened to the company. "It was awful," he says, "they said we were unsuitable and our support was unwanted. They were losing millions of dollars and instead of being grateful for our show of confidence in Sotheby's future, they did what they could to insult us. If Llewellyn had taken a course in public relations, he would have gotten an F."

Angry at being spurned and determined now to protect their already considerable investment, Swid and Cogan began devising a strategy for an all-out takeover. As rumors of their intentions circulated, the price of Sotheby's shares edged higher. When they finally put forth their bid April 11, it was for almost $100 million. A Sotheby's press release said the offer was "wholly unwelcome." A letter was produced, signed by 133 of Sotheby's professional specialists who threatened to seek "alternative employment" if the deal went through.

Cogan and Swid set up headquarters in a Mayfair hotel suite. They put out a prospectus promising to retain the traditional character of the company and they enlisted as one of their potential directors Lord Harlech, who as David Ormsby-Gore had been Britain's ambassador to the United States during the Kennedy administration.

Nearly 50 percent of the Sotheby's shares were already thought to be in American hands, and those shareholders therefore considered susceptible to the high price being offered for them. But, said Swid in an interview, "English investors were also liquidating to us like it was Dunkirk. Obviously they want to take advantage of a generous price."

Specialists in the auction world reported in the British press that some of the Sotheby's experts were reassessing their earlier hostility and saying they had acted under pressure in signing the petition. While the first deadline for the transaction is tomorrow, a consensus has already emerged that Swid and Cogan will finally prevail. The sale could be delayed by a government probe of whether Sotheby's transfer to American management represents an unacceptable loss of a British national asset, but that is thought unlikely.

Facing defeat, members of the Sotheby's board have nonetheless continued to counterattack. In a "defense document" board members argued that the company was on the way back to profitability and contended that Swid and Cogan had borrowed heavily to raise the money for their bid and would, therefore, be burdened with debt. Also noted in a section on the Americans' suitability to own Sotheby's was the fact that Cogan had some trouble with the Securities and Exchange Commission in the early 1970s and signed a consent decree that, while not conceding any wrongdoing, disqualified him from managing certain brokerage accounts.

The board members predict publicly that Sotheby's name will be degraded, franchised out as a label on furniture or a brand of cigarettes. All this, they say, represents, a violation of the credo of fine arts auctioneering, "which stands at the intersection of the business world and the worlds of scholarship and connoisseurship," as the defense paper asserted.

None of this will happen, says Swid. "Why would we go to all this trouble to kill the company?"