Secretary of State Alexander M. Haig Jr. clashed head-on with the White House and lost yesterday in a highly visible struggle that damaged his prestige and may yet lead to his resignation.

Haig was reported to be dispirited by the outcome of the day's developments and waiting to see what could or would be done today to shore up his authority before deciding whether his usefulness as secretary of state is at an end after only two months in office.

The bureaucratic struggle surfaced when Haig went public early yesterday with complaints about foreign policy management in the Reagan administration. By nightfall, he was put in his place by presidential order. The White House action, telephoned to reporters about 6:15 p.m., said Vice President Bush had been named to coordinate and control governmental action in time of international or domestic crisis.

Early in the day, Haig said that he had read with "a lack of enthusiasm" newspaper reports that Bush was being given such a role.

"I don't think a decision has been made on this issue; at least it has not been discussed with me if one has been made," Haig told a House subcommittee.

If a decision had been made without his being told, "that would pose another set of problems," Haig added.

The initial White House reaction to Haig's testimony, including his statement that he is "not totally satisfied" with the policymaking procedure, was to duck a fight.

In mid-afternoon, press secretary James S. Brady sought to minimize any disagreement. He had no criticism of Haig, and said an announcement on crisis management, which earlier had been promised this week, might be delayed.

In the Oval Office shortly thereafter, President Reagan and his advisers met to discuss the now highly visible problems raised by Haig's testimony.

The result was the White House statement announcing that Bush indeed would be crisis coordinator. The five-paragraph announcement said Reagan's choice of Bush "was guided in large measure by the fact that management of crisis has traditionally -- and appropriately -- been done within the White House."

"As in the past, the National Security Council staff will provide the administrative and other staff support to the president and vice president for the crisis management team," said the announcement, which made no mention of Haig or the State Department.

According to White House sources, Haig was informed of the decision by the president in a telephone conversation. A White House official said he understood that Haig was "definitely on board" after the call from Reagan. The White House denied a broadcast report that Haig had threatened to resign.

The day's events laid open to a greater degree than ever the crosscurrents of internal politics between White House officials and the assertive secretary of state. At day's end, Haig was clearly the loser.

A senior presidential aide said it had been considered "important to close the loop" on crisis management because of the publicity.

The official said there is a shared belief that "more close coordination" and "more communication" between Haig and the White House are needed, and expressed the belief that this can be accomplished by new awareness and improved procedures.

Among the patch-up efforts made known at the White House are frequent participation by Haig in the president's early morning national security briefing and more frequent meetings on other occasions among Haig, Reagan and presidential advisers.

The proud and combative former general, who billed himself only yesterday as the "general manager" of foreign policy in the Reagan administration, closeted himself for several hours last night with close aides to ponder the situation.

Haig has staked out a highly visible claim to be the man in charge of foreign policy second only to the president, and sometimes the president's role seemed an afterthought. A Time magazine cover earlier this month depicted a flinty-eyed Haig, hands on hips, blazoned with the words, "Taking Command." To some presidential aides, this symbolized the danger that the secretary of state might upstage his boss and outmaneuver the other actors in the policymaking process.

Such concerns, according to informed sources, generated a series of actions and leaks to the press in recent weeks to cut Haig down size. Haig, faced with increasingly obvious White House staff resistance, became increasingly uncomfortable.

Some Reagan aides, aware of Haig's brief and abortive bid for the 1980 Republican presidential nomination, were wary that the former general still harbors ambition for the highest office.

A source close to Haig, on the other hand, described Bush's selection as crisis manager over Haig as the first skirmish of "post-Reagan" politics.

A senior administration source said some people in the White House have been uneasy since the start of the administration. They were depicted as seeking to make certain that this is "a team approach" involving "an equalizing" of the various members.

The basic problem was that in experience, visibility and the view of the outside world the others were not equal to Haig, the former White House chief of staff and four-star NATO commander calling himself "the vicar" of foreign policy. In recent days, especially since a spate of newspaper and television reports last weekend, Haig expressed growing irritation to intimates.

This was well-known among the top ranks of the White House staff and State Department, but there was considerable surprise that Haig chose to make his views public. This nearly guaranteed that the White House team would have to meet the challenge.

Senior White House officials, in a tone of puzzlement more than anger, described the disagreement about crisis management as a misunderstanding. It was clear from the first that international crisis would have to be run from the White House, as in previous administrations, they said.

There is considerable precedent, including that of Zbigniew Brzezinski in the Carter administration, for the president's national security affairs adviser to be the crisis coordinator.

This was not proposed this time, according to White House insiders, because of a desire to maintain a low-profile role for current national security adviser Richard V. Allen lest he be considered a public competitor to Haig.

Consideration as crisis coordinator apparently was given to presidential counselor Edwin Meese III or chief of staff James A. Baker III, who were described by one source as "natural candidates." However, the job was seen as a more logical function for Bush, whose high post as the nation's second-ranking elected official would seem to minimize suggestions that Haig had been bypassed.

All of this was discussed with Haig last week, a White House official said. It was thought that Haig understood that naming Bush would be the course most in Haig's interest, the official said, and that Haig agreed.

The mistake, as seen from the White House, was that Reagan had not spoken directly and finally to Haig about the decision to designate Bush as crisis manager.

This was quickly remedied in yesterday afternoon's telephone call, but only at the cost to Haig of considerable prestige and considerable disarray to the Reagan administration as a whole.

"The thinking here is that Al is doing a superb job," said a White House official involved in yesterday's encounters. "A lot of this is start-up problems. We haven't had as much in the domestic arena, where there where there isn't so much natural pulling and tugging" on policymaking, he said.

Haig is extremely conscious of his turf and sensitive to any doubts about his authority, possibly because he watched his then-boss, Henry A. Kissinger, gather most of the real diplomatic power in the Nixon administration, cutting out then-secretary of state William P. Rogers.

In recent days, Haig has privately expressed unhappiness that he and State did not seem to have the leading role in such internationally affected policy issues as the grain embargo against the Soviet Union, the effort to protect the American automobile industry against Japanese imports and immigration and refugee matters.

Even for the hard-charging secretary of state, yesterday was a killer, in pace if not in substance.

He began with a hastily scheduled 7:30 a.m. meeting with the visiting Japanese minister.

By 9:30 a.m. Haig was on Capitol Hill to testify before the House Foreign Affairs subcommittee on international affairs about the State Department budget.

While volunteering that Reagan had indicated that Haig would be "the general manager of American foreign policy," Haig sounded less confident than before, even dispirited.

After 75 minutes of testimony, Haig went to the White House to attend a meeting with Bush and Japanese Foreign Minister Masayoshi Ito.

Back at the State Department, he lunched with Soviet Ambassador Anatoliy F. Dobrynin.

Haig was reported back at the White House for meetings during the afternoon, before a session with Mexican Foreign Minister Jorge Castenada and a meeting with British Ambassador Nicholas Henderson.