Federal Reserve Board Chairman Arthur F. Burns made no secret of it. Although 73 and with 11 years of government service behind him, he very much wanted to be reappointed to his command post at the U.S. central bank.

He would be gratified, he told friends, if the President gave him another four-year term as chairman. At the same time, he recognized that the odds were against him, and he said that he would greet the resolution of the uncertainty with some relief.

It was also no secret that President Carter's entourage of economic advisers, if not Carter himself, had grown increasingly disenchanted with Burns. It was not so much the image of Burns as a conservative Republican - although Burns likes to refer to himself as "old fashioned" - as the feeling that Burns refused to be a team player.

"It's not just a question of giving bad marks on Burns' monetary policy,' one prominent Democratic economist said yesterday. "Burns had opinions that he freely offered on everything, including fiscal policy, and the town just wasn't big enough for him and Carter."

Such Carter advisers as White House aide Stuart Eizenstat and Council of Economic Advisers Chairman Charles L. Schultze argued that with

Burns at the helm of the Federal Reserve there could be no coordination on fiscal and monetary policy.One arm of the government would go one way, they said, while Burns would go the other way.

The entire Carter establishment soured on Burns for his successful-one-man campaign to kill the $50 tax rebate originally proposed by the President as part of his economic stimulus package.

But the dilemma for Carter was Burns' extraordinary standing with the financial community abroad and businessmen at home. He was Mr. Fiscal Integrity personified.

Early on, the Carter team held the notion that retention of Burns might be indispensible to maintenance of business confidence in the new Carter administration. But Eizenstat and Treasury Secretary W. Michael Blumenthal argued privately that the confidence of businessmen - or to put it better, their respect - would depend on Carter's performance. And Carter's ability to keep the economy moving and growing, they feared, they would be inhibited by Burns.

In a midsummer speech, Schultze pointed out that the economy's modest economic growth rate was threatened by a sharp upturn in interest rates that was being deliberately engineered by the Federal Reserve.

If this trend continued and affected long-term interest rates, which relate to housing and business investments, Schultze warned that recovery could stop and recession could begin again.

But in speech after speech, Burns made it clear that his main concern was the threat of a potentially dangerous inflation which, he claims, could bring on recession. Under his guidance, therefore, monetary policy continued about unchanged, although there was some evidence coming to light that Burns no longer had a unanimous board behind him.

Meanwhile, during the September session of the International Monetary Fund annual meeting in Washington, prominent bankers passed the word that they no longer felt it essential to keep Burns in office, inasmuch as Carter - once unknown to them - had proved himself to be a moderate on economic issues.

In conducting a talent search for a replacement, Vice President Mondale took as his guide the replacing of Burns with a man who would be satisfactory to business, but who had a social conscience as well. Miller, as Chairman of the National Alliance for Businessmen, is reportedly concerned with problems of long-term unemployment.

He was especially recommended by Blumenthal, and won support, before the appointment, from Reginald Jones, co-chairman of the influential Business Roundtable. In the words of a close Carter intimate, they sought to find "someone as good as Burns."

Burns, a brilliant economic theorist, never tried to shield his strong opinions, or make his convictions conform to political necessity. Many though him highly opinionated and stubborn, but no one ever challenged his intellectual intergriitiyi. He was at his very best in testimony before Congress, where he won many friends on both sides of the aisle.

He was a skilled tactician as well, fending off for many years all congressional efforts to circumscribe the enormous power of the Federal Reserve System. But in the past few years, he has been losing ground, especially to Congress' demands that he establish money growth targets and report on it in regular appearances before the House and Senate Banking Committees.

He was facing a further effort, through the Humphrey-Hawkins "full-employment" bill; to say how he would adapt monetary policy to economic objectives set out by the President. Burns' view is that this would make a mockery of the so-called independence of the Fed.

If he stays on the board, his friends say, it would not be out of annoyance or pique, but because of devotion to the Fed as an institution and the belief that his presence on the board would be important to world economic stability.

If Burns decides not to stay on as a member of the board, it would enable Carter to appoint another new member, and the assumption is that he would designate a liberal to balance things out.