Duke University has a big problem with its most famous alumnus. It keeps his portrait hidden in a vault. His name doesn't appear anywhere on campus. And when trustees tried to give him an honorary degree, the faculty rejected it.

A great many people here simply would like to forget that Richard M. Nixon ever graduated from the Duke Law School.

But now many fear the university's name and reputation will become permanently linked to Nixon by construction of a Nixon presidential library on university property. Duke President Terry Sanford wants such a library, and has staked his prestige on bringing it to the campus.

The proposal, which is to go before university trustees this week, has touched off a virtual rebellion among some of the university's most respected faculty members and alumni.

Some of the most outspoken words have come from the history and political science departments, which supposedly would benefit most by such a library. "We know more about presidential papers than Terry Sanford," said history professor Lawrence C. Goodwin. "We know what presidential libraries are. They're not archives; they're shrines."

But harsh words also have come from elsewhere. "To build a library here is an effort to recognize a man who disgraced the presidency and dishonored his country," said H. Sheldon Smith, professor emeritus of American religious thought. "It would be an albatross around Duke University for years to come; an object of censure, scorn and derision."

The controversy, which has split the university more than anything since the days of student unrest in the 1960s, is full of ironies. The most interesting involve Sanford.

A former North Carolina governor, he is among the nation's best known university presidents, a man with a secure political and academic reputation, a liberal in a conservative southern state. His credentials as a Nixon opponent are impressive. At considerable political risk, he seconded the nomination of John F. Kennedy at the 1960 Democratic convention and in 1972 launched an ill-fated attempt to win his party's nomination to oppose Nixon.

The most persistent question asked here these days: Why is a man like Sanford risking so much for Nixon?

"The neck of the president is at stake," said Smith, who came here from Yale 32 years ago. "Why would a man who has a fine record jeopardize his future and the university? You might have to say this man has lost his perspective and is insensitive to what is at stake."

Sanford, at 64 and in the twilight of his presidency, argues there is no risk to himself or the university in the library conflict.

"I would have been ashamed of myself if I would have been afraid to propose this," he said. "I think it indicates integrity, courage and institutional self-confidence on Duke's part. I think it would hurt our image if we turned it down.

"These are the most extensive, controversial and probably most interesting presidential papers in history," said Sanford. "More timid people might shirk from controversy. But I feel the opposition of the moment will be overcome by the long-range benefit to scholarship and that's what a university is all about."

Sanford first approached Nixon about the possibility of a presidential library at Duke during a meeting in New York on July 28. Nixon was amicable, but noncommittal. The two men joked about the prospect of Duke hiring Nixon's old friend, George Allen, the former Washington Redskins coach, to take over its football program.

But within the next 10 days, Nixon's lawyer, R. Stan Mortenson, had visited Duke twice. Nixon was interested.

He apparently has warm feelings about Duke. He entered its law school on a $250-a-year scholarship during the heart of the Depression, and was well liked and respected as a student. He worked in the law library, served as president of the student bar association, and in 1937 graduated third in his class. He has regularly contributed to the law school alumni fund ever since.

Like many other students, Nixon was dirt poor. With three classmates, he lived in one large room in a rundown house, called Whippoorwill Manor, with a woodstove and no running water. He seldom socialized, and was nicknamed "Gloomy Gus" by his classmates.

The campaign for the Nixon library began secretly. Sanford first sold the idea to top university administrators and a select group of senior faculty members. They were told a group of Nixon's friends, led by former ambassador Walter H. Annenberg, would raise the $25 million needed to build the library. Duke would simply donate the land.

The first hint of opposition didn't come until Sanford phoned Richard L. Watson, acting chairman of the history department on Aug. 8. Watson was shocked. His anxieties rose when he was told later that the matter had to be settled by Aug. 19 because the University of Southern California was concluding lengthy negotiations for Nixon's papers.

"All that is false," Cornelius J. Pings, senior academic vice president at USC, said. Although negotiations between USC and Nixon began in 1977, "relatively nothing" has occurred between them in years, he said.

Watson rounded up 13 members of the history department for a meeting. Twelve of those present, including two conservative Republicans, opposed the library.

"With few exceptions our concern is that the possibility of dramatically acquiring a magnificent presidential library may obscure the aura of dirty tricks which will inevitably surround this particular presidential library," Watson wrote Sanford in explaining the department's position. "We'd all love to have Benedict Arnold's papers, but we don't want a Benedict Arnold building on campus."

Opposition has been growing ever since. There have been endless meetings and strategy sessions. Countless arguments. Hundreds of "information" sheets circulated. Much hand wringing.

The key players aren't a bunch of students in blue jeans or radical young professors. They are gray-haired department chairmen and scholars with well-established reputations, men and women not given to quixotic adventures.

Pulitzer-prize winning author William Styron, a 1947 Duke graduate, has issued a letter attacking the library. Fourteen of the university's most respected professors signed a similar broadside. One trustee emeritus, Charles Murphy, a Washington lawyer and one-time adviser to Harry Truman, resigned from a host of alumni positions.

And last Thursday, the political science faculty voted to oppose the library unless a decision on it can be delayed.

Part of the opposition centers on Sanford's apparent attempt to ram the library through while most of the faculty and student body were away for the summer.

But there are dozens of other questions circulating on campus:

Who will control the library? Actually what will be in it? The Nixon tapes? Pictures of Spiro Agnew? Will it be an archive for presidential papers, or a monument to the Nixon presidency? What would Duke gain by having it on campus? Money? How often would it bring Nixon to Durham? Is Nixon, in a desire to get a library somewhere, playing Duke off against USC? Would Duke, in effect, be endorsing the Nixon presidency by donating land?

Most boil down to a question of image: Would a Nixon library damage Duke's reputation?

Almost everyone would love to have the former president's papers at Duke, if only they could be quietly put in the university library. Some, like law professor Walter Dellinger, think even broaching the subject with Nixon was a mistake. "I fear the university in entering negotiations with Nixon landed a strong punch in the belly of a tar baby," he said.

Sanford has handled the matter like the skillful politician he is. He has been both flexible and forceful. In an attempt to pacify faculty opponents, he agreed to delay a decision on the library until Sept. 4, the day after the Academic Council, which represents the faculty, has scheduled a vote on the library. The faculty will debate the issue today. (Classes don't begin until after Labor Day.)

In an attempt to build alumni support, he mailed 65,000 letters outlining his position. Included with it was an letter from English professor Edwin H. Cady calling the library "a superb idea," but no word from opponents.

"We all got the idea that a very big train was coming down the track," said one professor.

The library has many powerful and articulate supporters on campus. Reynolds Price, a novelist and English professor, for example, thinks the moral outrage among his colleagues is misplaced.

"I don't think the existence of a library is an endorsement of a career," he said. "Europe would be an empty place if all the buildings bearing the names of morally apprehensive characters were torn down . . . . Nobody goes to Grant's Tomb and shudders at the evils of the Grant administration."

Law School Dean Paul D. Carrington said the Nixon papers "may be the most valuable collection of 20th century documents. The likelihood is there's some really good stuff in there."

Nixon is a "loyal alumnus," he added. "We are associated with him. He is one of us. We don't have any choice in that."

Paradoxically, Carrington said he thinks it "premature" to take the law school's portrait of Nixon out of the vault where its been kept since the height of the Watergate controversy. "We're still in a time when someone might deface it," he said.