British Foreign Secretary Francis Pym, stressing that "there is always hope, but I have never disguised the difficulty," met with Secretary of State Alexander M. Haig Jr. for almost four hours yesterday in search of ways to avert warfare between Britain and Argentina over the Falkland Islands.

Pym, who came here to convey new British proposals for a peaceful resolution of the crisis, called the meeting "a useful start" and said he and Haig would resume their talks today. But, he told reporters after yesterday's session, "There is much work to be done. It is a very difficult problem."

The foreign secretary, who will meet with President Reagan this morning and who tentatively plans to return to London tonight, declined to discuss specifics of the ideas being put forward by Prime Minister Margaret Thatcher's government as a counter to an Argentine plan rejected by Britain.

His visit came as U.S. officials cautioned that time was running out on Haig's two-week mediating effort. The British naval force, dispatched by Thatcher after Argentina's forcible occupation of the Falklands on April 2, is approaching striking distance of the South Atlantic islands, making more imminent the possibility of an armed clash that would force the United States to choose sides.

British sources accompanying Pym said without elaborating that press reports circulating in London about details of the British plan were not accurate. They added that, while Britain wants to be flexible and is willing to consider a "wide variety of permutations and combinations that might go into a solution," London will not accept any formula inconsistent with the three principles on which it has insisted since the start of the crisis.

The sources reaffirmed these principles as withdrawal of all Argentine forces from the Falklands, reestablishment of British administration over the islands, which Britain has controlled since 1833, and negotiations for a long-term solution, with the wishes of the island's 1,800 residents being the paramount consideration.

While that framework appeared to show little give in the British position, the sources said they were hopeful that room for compromise might be found in fleshing out the specific details. To the degree that they would elaborate, the sources seemed to indicate that the Thatcher government is unyielding on some points but disposed to give ground on others.

On the question of interim administration following an Argentine withdrawal, one source said it would have to be identifiably British, although he conceded that London is not disposed to get "too hung up" over what flag flies over the islands.

That appeared to hint at British attempts to meet Argentina's emotional insistence that its sovereignty over the Falklands, which it calls the Malvinas, be acknowledged. The Argentine plan, conveyed to Haig in Buenos Aires earlier this week, had proposed an interim administration period; however, it added that the transition should be very brief and result in the "full exercise" of Argentine sovereignty.

In respect to a long-term solution, the sources said Britain is willing to negotiate all questions, including that of sovereignty. But, they added, Britain insists that there be no prejudging of the outcome of any negotiations and that the islanders' wishes be given priority consideration.

Press reports from London have said that Thatcher is willing to concede, at least tacitly, the sovereignty issue, and some of these reports have speculated that she might be willing to try to convince the islanders to accept some form of Argentine rule. While Haig was meeting with Pym, State Department spokesman Dean Fischer said the secretary plans to meet at some point with Argentine Foreign Minister Nicanor Costa Mendez, who is expected here late Saturday. Costa Mendez is coming for a special meeting of the Organization of American States scheduled for Monday to consider a possible Argentine request for assistance against Britain.

Defense Department spokesman Henry Catto said yesterday that a U.S. Navy tanker carrying jet fuel was nearing Ascension Island in the Atlantic between Africa and South America to replenish supplies being used by planes of the British task force. It was the second U.S. fuel shipment to the island since the crisis began. Although the United States has tried to maintain an evenhanded approach to facilitate Haig's mediation efforts, Britain has the right, under a 1962 agreement, to use facilities on Ascension. The island lies about midway between Britain and the Falklands, and Britain has used it as the resupply point for its task force.

In the meantime, growing signs of impatience with the U.S. stance of evenhandedness and demands for open support of Britain continued to be evident in Congress. Sen. Paul E. Tsongas (D-Mass.), a member of the Foreign Relations Committee, which met with Pym yesterday, said earlier that the Reagan administration should "condemn unequivocably, openly and loudly" Argentina's resort to force. Tsongas spoke a day after Sen. Daniel Patrick Moynihan (D-N.Y.) called on Reagan to impose a trade embargo against Argentina.