Secretary of State Alexander M. Haig Jr. told President Reagan yesterday that Britain is in position to win an early victory over Argentine forces in the Falkland Islands, and administration officials are understood to believe that British recapture of the islands is likely to be complete within 10 days.

Following a White House meeting with Republican congressional leaders, House Minority Leader Robert H. Michel (Ill.) quoted Haig as saying: "The British appear to be in a position militarily to bring the war in the Falklands to an early conclusion."

State Department spokesman Dean Fischer later confirmed that the quote was accurate and said it represented Haig's "assessment of the situation."

These views were expressed before reports later in the day that at least one British ship had been badly damaged by Argentine forces. But those reports did not change U.S. officials' general assessment that is based on the number of British troops landed on the islands and Argentina's dwindling numbers of aircraft and trained crews.

However, while senior U.S. officials believe that the short-term military aspects of the seven-week crisis are about to be resolved in Britain's favor, they see little immediate chance for a long-range agreement on the future of the disputed islands that would prevent further conflict.

Haig and other top administration officials are known to be probing for opportunities to coax Britain and Argentina into renewed negotiations on the sovereignty question.

Haig is known to have sent a message to British Foreign Secretary Francis Pym cautioning that insistence on an all-out military victory that humiliates Argentina could stir resentments not in the interests of the West.

In the text of an interview given to European journalists Monday and made public here yesterday, Haig also alluded publicly to this U.S. view. He said: "It's been a tradition of Great Britain when in conflict to be vigorous, but in victory to be magnanimous. I would hope that pattern established by Winston Churchill will continue."

However, the U.S. efforts have encountered new inflexibility by Britain and could be impeded further if the anticipated Argentine military defeat provokes a political upheaval there.

According to U.S. sources familiar with the situation, Prime Minister Margaret Thatcher's British government, after making a costly investment in lives and materiel to recapture the islands, is unwilling to reconsider concessions it originally offered in hopes of averting warfare.

Instead, the Thatcher government is described as leaning toward keeping an interim British administration over the Falklands for approximately two years before seeking to place them under a United Nations trusteeship that would preserve British citizenship of the 1,800 inhabitants.

This tentative British plan would reject recognizing Argentine sovereignty and would permit Argentina only minimal commercial rights in the Falklands and surrounding waters.

The administration is understood to believe thatArgentine President Leopoldo Galtieri and members of the ruling military junta are unlikely to survive the failure of their Falklands gamble.

In the U.S. view, the armed forces seem certain to retain their control of the Argentine government for the foreseeable future. But, if Galtieri and the junta are ousted, a prolonged period of political chaos and revolving-door governments could ensue, with different factions of officers competing and playing on Argentine bitterness with promises of a future attempt to win the Falklands.

If that proves to be the case, administration officials see a danger that Argentina's sense of humiliation will find sympathetic echoes in other Latin American countries and cause new tensions pitting the Hispanic world against Britain and its "gringo" allies such as the United States.

At Argentina's insistence, foreign ministers of the Organization of American States are to meet here Thursday to consider what could turn into a new Argentine request for aid under the 1947 Rio treaty of reciprocal hemispheric defense.

At the very least, U.S. officials are concerned that the session will produce impassioned pro-Argentine oratory with increasingly anti-American overtones.

While it is not clear whether Argentina will seek more than rhetorical support, U.S. officials doubt that Buenos Aires can obtain enough outside help to affect the military situation.