Livingston L. Biddle Jr., an aide to Sen. Claiborne Pell (D-R.I.), has emerged as the front runner for the government's top arts job.

Knowledgeable sources here say Biddle, 58, will succeed Nancy Hanks as chairman of the National Endowment for the Arts, which has an annual $123.5-million budget, when her second four-year term expires Oct. 3.

White House spokesman Barry Jagoda says that President Carter has not yet begun "to focus on the issue." But people in the government and in the arts constituency who claim to know his thinking and the politics and pressures guiding his decision feel, as of this writing, that Biddle is the one.

Time is running short. Once the President nominates a candidate, the name must be submitted, for a back-ground check, to the FBI. That will take at least a month. Then the nominee must win the approval of the Senate. That also will take time. If a new chairman's name is not sent to the FBI by, say, mid-September, Senate confiramtion will be postponed to the next session, perhaps until March.

"The procedure at the White House for filling such a post begins with the personnel department," says Jagoda. "They'll send a memo to the President through Hamilton Jordan, advising him of the situation. That hasn't happened yet. The President, I'm sure, also will be seeking Joan Mondale's advice."

Biddle is no artist, though he was written novels: nor is he thought ambitious for elective office; nor, though he has raised money for the Pennysylvania Ballet, is he known to favor one art or another. These factors, although negative, all work to his advantage.

Biddle's skills are tecnhical. He is a professional arts bureaucrat.

In 1953, as an aide to Pell, Biddle drafted legislation that led to the establishment of, first, the National Council on the Arts (1964), and then the National Endowment (1965). In 1966, when the agency was funded (its initial appropriation was only $2.5 million Biddle was appointed its first deputy chairman.

In 1968, Biddle left the government. He spent three years at Fordham University, establishing a new liberal arts college, and two as the board chairman of the Pennsylvania Ballet, before returning to this city to work again for Pell. He has served as NEA's congressional liason director. He knows Hill staffers, the congressmen they serve, the bureaucrats they deal with, and the growing number of powerful arts lobbyists who put pressures on them all. Last year he was named the staff director of Pell's subcommittee on education, arts and humanities.

Other candidates for the $52,000 job have been rumored, in recent months, to include:

Nancy Hanks, who is, everyone agrees, a hard act to follow. She is a woman, she's effective, but she also is a Republican, long associated with the Rockfellers, and she has had eight years in the job. There are Democrats on the Hill who disapprove of how she has personified the agency (some of them cite J. Edgar Hoover and the FBI). Eight years is enough, they say, Democrats have long voted funding for the arts. There is a new administration, let's give a Democrat the job, They add.

Wes Uhlman, the mayor of Seattle. Uhlman, though a Democrat and a supporter of the President, has slimmer arts credentials. "Uhlman scares the hell out of us," says one prominent arts lobbist. "We hear that he still wants to run for office back in Washington. He's looking for a holding pattern. Biddle is not running for any other job."

Michael Straight, the Endowment's deputy chairman, is another Republican appointee closely associated with Hanks. He does not expect the post. "I think my days are numbered here," he says.

Peggy Cooper of Washington, founder of the Workshop for Careers in the Arts, who is bright, female, black and young - too young, say her critics. Sources say her chances have diminished in recent months. Perhaps she thinks so, too. At any rate she's taken a new job with PBS.

As their chances dim, Biddle's prospects brighten. "Who else is there?" asks Peter Zeisler of the Theater Communications Group, New York. "If the arts in this country were healthy, we'd have seven or eight candidates on the tips of our tongues. And we don't."

Biddle acknowledges that if he gets the job, it "may be by default."

"We don't want to wait until next March," says one well-known "arts advocate." Michael Newtown, president of the American Councils on the Arts. "All we are asking, Mr. President, is please send up a name before the end of October."

Sources who suppose Biddle a front runner say that he must first cross a major hurdle - that evoked by his artistocratic name.

Livingston Ludlow Biddle is a Biddle of the Philadelphia Biddles. (The first two novels that he wrote were called "Main Line" and "Debut.") An uncle, Anthony Drexel Biddle, was they say, the model for the title role of the play "The Happiest Millionaire." Two of Biddle's cousins, Anthony J. Drexel Biddle and Angier Biddle Duke, were appointed U.S. ambassadors to Spain.

At St. George's prep school, and later at Princeton, Livingston Biddle was a classmate and close friend of his most important champion, the Rhode Island politican who is sometimes called Sen. "Wellborn" Pell.

Pell, the Hill is well aware, blocked the reappointment of Ronald Berman to the chairmanship of the Humanities Endowment on the partial grounds that Berman, the Ford administration choice, had proved himself "elitist.

That word is often used, if infrequently defined, by campaigning politicans. During the last campaign, when speaking of the Humanities Endowment, Jimmy Carter used it, too.

"What can I say?" Biddle asks. "My name is Biddle, true, but I developed my own philisophy during World War II." (He drove an American Field Service ambluance on the battle fields of Europe.) He is sitting in his tiny Capitol Hill cubbyhole surrounded by books, bills papers. "Perhaps the idea of serving other people has been ingrained in me since childhood."

Biddle knows the FBI has not begun checking on his background. He also is aware that being spotlighted as the leading candidate may not help his chances.

He says only friendly things about Nancy Hanks and her stewardship of the Arts Endowment. "The arts," he says, "are now far better organized then when she took the job. The advocacy organizations, the opera people, and the people who support museums or the theater, and the state and community people have to work with one another if the federal budget for the arts is too significantly increase. Special interest groups, defeat one another's efforts. They should form a united front. The issue isn't mass versus class. You have to maintain quality. I do not know who will get the chairmanship, but I do know that making quality available to the maximum number of citizens in the essence of the job."