Secretary of State Alexander M. Haig Jr. and British Foreign Secretary Francis Pym ended two days of talks last night with Haig getting a cautious go-ahead from Britain to approach Argentina with new U.S. ideas for averting a clash over the Falkland Islands.

Shortly before Pym returned to London, a well-informed British source said the talks had produced a document that does not have the formal approval of Prime Minister Margaret Thatcher's government and that is "not satisfactory" to the British.

He added, though, that the document, whose contents he would not describe, might produce a basis for continuation of Haig's efforts to mediate between London and Buenos Aires.

The source insisted that the document bears "no Anglo-American label" and consists solely of U.S.-proposed ideas that, in the British opinion, "still leave a great deal to be desired."

"The ideas are from the British view unsatisfactory. The question we the British have to address is: are we willing to live with them as something to be taken forward by the Americans for further negotiation?" he said.

That decision will be made by Thatcher after she hears Pym's report today, he added. But, in response to questions about whether Haig is being encouraged to pursue the U.S. ideas with Argentina in the interim, the source replied, "We hope his mission will continue."

His ambivalent approach to the latest turn in the three-week Falklands crisis appeared to be part of a delicate diplomatic ploy aimed at keeping the Haig mission alive without putting the Thatcher government in the position of tying itself to proposals that might prove unacceptable in Britain.

The source seemed, in effect, to be signaling that Britain wants Haig to keep trying but reserves the right not to be bound by any of the U.S. ideas in the latest U.S. proposal and to repudiate them if it sees fit later.

In contrast to the carefully hedged British attitude, U.S. officials were understood to have left the meetings optimistic that they have gained additional maneuvering room that will permit Haig to keep the crisis, for the time being, from exploding into warfare.

U.S. sources said last night that they view the ideas hammered out by Haig and Pym as a "first version" of a proposed solution which will be unacceptable in its entirety to Argentina but which contains sufficient enticements to engage the Argentines in renewed, serious bargaining.

The sources said Haig will present the ideas to Argentine Foreign Minister Nicanor Costa Mendez, who is due here for a meeting Monday of the Organization of American States. They said the two probably will meet Sunday, or, if Costa Mendez arrives early enough, tonight.

However, the U.S. sources noted that final decisions about Argentina's position must be made by President Leopoldo Galtieri and the generals dominating his military regime.

For that reason, they said, Haig's expected meeting with Costa Mendez is regarded as a conduit for transmitting the U.S. ideas to Argentina and, if their reception is favorable, paving the way for resumption of Haig's shuttle diplomacy in Buenos Aires within days.

The U.S. sources also were reluctant to describe details of the latest proposals. They did say that, despite British insistence on describing them as purely American in origin, they cover specific ways in which Britain is willing to show flexibility.

The chief sticking points involve Argentina's demand for recognition of its sovereignty over the Falklands and Britain's insistence that, before any negotiations for a long-term solution, Argentina must remove its forces from the islands and reestablish British administration.

According to the sources, the latest proposals are designed to address these problems through new variations on previous proposals such as permitting flags of both nations to fly over the islands and providing a relatively brief interim administration before determining the Falklands' ultimate status.

The U.S. sources also said that, while the situation is too volatile to make predictions with certainty, the results of the Haig-Pym talks had made them more optimistic that there will be no immediate outbreak of fighting in the South Atlantic.

For that reason, they said, such developments yesterday as an Argentine report of two British warships approaching South Georgia Island about 800 miles from the main Falklands and Pym's repeated insistence that Britain will not commit itself to refrain from military action appeared to be maneuvers in a war of nerves rather than an omen of imminent warfare.

The carrot-and-stick tactics being used by Britain were typified by Pym's remarks in an interview yesterday with the British Broadcasting Corp.:

"We remain determined to work for a peaceful solution if it is humanly possible. But we shall not shrink from the use of force if that is necessary, and it would be no service to the cause of peace for us to pretend otherwise."

He delayed his return home on a Royal Air Force jetliner by two hours to have a final meeting with Haig last night. Before the departure, Pym's spokesman, Nicholas Fenn, told reporters, "There is life in the Haig mission, and he Haig will do everything humanly possible to find a peaceful solution."

The two days of discussions were marked by questions from reporters about why Pym had not met with President Reagan. Fenn said a report Thursday that such a meeting had been planned did not come from the British delegation.

He added that no meeting with Reagan had been requested or planned and that Pym's meetings with Haig, and a breakfast session yesterday with national security affairs adviser William P. Clark, were adequate to completion of his mission.