Under heavy American pressure, Taiwan has abandoned demands that its U.S. relations continue on a government-to-government basis and has agreed that future ties be handled by private organizations staffed by nongovernment employes.

But, in a concession to Taiwanese sensibilties, the Taipei government will be alowed to say that it considers its relations with the United States to have the "quality of officiality."

Reliable sources said yesterday these are the key points of the deal that has been worked out to handle the sensitive question of future U.S.-Taiwanese relations in the wake of President Carter's decision to break the three-decade tie with Taipei and recognize Peking as the government of China.

In line with Peking's insistence that Washington cannot have any official dealings with Taiwan, the administration called for creation of private institutes to handle the trade, cultural and consular matters previously administered by the embassies of the two countries.

According to the sources, the administration's position has prevailed despite a vigorous lobbying campaign by Taiwan to win support in Congress for some kind of governmental tie.

At a press conference yesterday, Carter repeated an earlier warning that he will veto any legislation that might "contradict the commitments we have made to the government of China."

Administration sources said the president's warning applies to possible congressional efforts to give Taiwan any official diplomatic standing in Washington or to other steps that might be regarded in Peking as going beyond the U.S.-Chinese agreement on normalization of relations.

Faced with this hang-tough attitude on Carter's part, reliable sources said, Taiwan reluctantly has agreed to follow Washington's lead and establish a private entity similar to the American Institute in Taiwan set up by the State Department last month.

According to administration sources, the United States will announce this week that Taiwan has agreed to the idea of nongovernmental bilateral relations and will create an office called the Coordinating Council for North American Affairs. Its organization will be similar to that used by the State Department for the American institute -- operating with government funds and staffed by diplomats temporarily separated from government service.

Taiwan has agreed not to contradict this American announcement, but it does not have to support it publicly. If asked, Taiwan will say that it regards the new arrangement as having an official character; the United States has agreed not to contradict this assertion.

Originally, the sources said, Taiwan had proposed what diplomats call "an agree to disagree formula" for overcoming the dispute about the nature of these organizations. Under that proposal, Washington would have been free to call them "unofficial" and Taiwan could have said they were "official." However, the sources continued, the United States refused to accept that idea, and Taiwan finally agreed to the U.S. contention that the organizations must be considered nonofficial.

In addition, the sources added, Taiwan will be allowed to make a distinction publicly between the character of these organizations and what it considers to be the nature of its U.S. relations.Specifically, it will be able to say that, since both organizations are financed by government funds, it regards U.S.-Taiwan relations as retaining "a quality of officiality."

However, it was not immediately clear whether this complex formula will enable the administration to head off efforts in Congress to give Taiwan a continued official status or other congressional expressions of U.S. support unacceptable to Peking.

Late last week, urgent lobbying by the administration apparently halted a proposed resolution being considered by the Senate Foreign Relations Committee. It said the United States would consider any use of force by Peking against Taiwan a direct threat to American security.

Although Congress is in recess this week, efforts continued yesterday to find compromise language that could be written into legislation authorizing funds for the new American institute.

Reliable sources said aides to several senators have produced a draft that would declare a continuing U.S. interest in Taiwan's security without stating the kind of explicit commitment that Carter has threatened to veto.

Although he reiterated his warning yesterday, the president for the first time indicated that he would accept a watered-down congressional statement on Taiwan. Previously he had dismissed such congressional action as "unnecessary," but he said yesterday: "I have never said I will not accept any resolution."

The president also elaborated on a weekend interview in which he said that he and future presidents retain the option of using American warships or other military means to protect Taiwan.

"I have no intention of going to war," he said. "I wanted to point out that no future decision by myself or my successor is prevented. But our country is one that believes in peace, and I have no anticipation that there will be any requirement for war in the Western Pacific."