Deputy Secretary of State Warren Christopher, saying that "time is running out," flew to Algiers last night in a last-ditch drive by the Carter administration to negotiate the release of the 52 American hostages in Iran.

Christopher's surprise trip followed by less than 24 hours the dispatch by cable to Algiers of U.S. answers to Iranian questions about the "mechanics and procedures" of Washington's latest proposals in the hostage dialogue. The indirect negotiations are being handled by Algeria, which has been designated by Iran as its authorized intermediary.

Several hours before Christopher's departure, he and Secretary of State Edmund S. Muskie met with President Carter to obtain approval of Christopher's instructions.

Christopher, who was accompanied by three aides, said upon departure from Andrews Air Force Base that "serious problems remain," but that the distance between Iran and the United States, while "still measured in large numbers, seems to be narrowing somewhat." Stressing that it only "seems to be" narrowing, he discounted the likelihood of an "imminent breakthrough."

In addition to providing personal explanations and suggestions in the effort to conclude the negotiations and bring about release of the Americans, Christopher may discuss the role of the Algerians in a potential agreement. In the past several days Iranian officials have suggested a greater role for Algeria in arranging and perhaps even guaranteeing a settlement of outstanding issues. Algeria is believed to be resisting a major expansion of its role.

According to the State Department announcement, Christopher will confer with Algerian Foreign Minister Mohammed Seddik Benyahia, whom Christopher saw on visits in November and December to convey U.S. positions to Iran. Benyahia, who is responsible for the operating instructions of the Algerian emissaries in Tehran, is considered an important behind-the-scenes figure in the effort to end the 14-month-long captivity of the Americans.

The State Department spokesman and officials dealing with the matter denied yesterday that an "agreement in principle" to accept American proposals has been transmitted to Washington by the Algerians. Such a report, by ABC News, caused a nation-wide flurry of speculation late Tuesday of an imminent breakthrough in the situation.

U.S. policymakers involved in the discussions said it is highly uncertain whether sufficient time remains to complet the deal in this administration. But the stepped-up pace of the U.S. efforts was clear evidence of Washington's determination to leave no possibility untried.

The most encouraging sign in the recent developments as seen by Washington officials is that, despite what Iran terms its "final position," offered Dec. 19, it is continuing the dialogue, and its questions suggest shifts in its position. But a discouraging fact is the rather large distance that remains to be closed in a very short time.

The Carter administration has told Iran that its proposals for a hostage settlement may be withdrawn by the incoming Reagan administration, and thus can only be considered valid U.S. commitments if a deal is completed by noon on Jan. 20. As a practical matter, the Carter administration said in an earlier message, it will be impossible to make all the arrangements in time unless Iran approves U.S. proposals by Jan. 16, next Friday.

President-elect Ronald Reagan said late yesterday that "i hope and pray" that the hostages are returned before he takes office.But if this is not done, he said, "I would want to look and see exactly what has been done. I don't want to write a blank check here."

The new series of diplamatic exchanges began at midafternoon Tuesday, when the State Department received a cabled message from the Algerians' emissaries containing questions about the most recent U.S. proposals.

A four-hour meeting of the U.S. negotiating team, which is headed by Christopher, at the State Department Tuesday night formulated the U.S. answers.

The U.S. answers, after being personally checked with Carter, were cabled to Algiers late Tuesday for transmission to the Algerian emissaries in Tehran.

A two-hour meeting of the Christopher group yesterday morning discussed the next moves in the dialogue. In mid-afternoon Muskie and Christopher went to the White House to brief Carter for a half hour.

Carter, at an earlier meeting over breakfast to say goodbye to a group of Georgia congressmen, was quoted as saying that no new development had changed the sense of frustration about the hostage issue that he has felt for many months. His breakfast guests said Carter expressed particular frustration with the continuing profusion of voices and division of authority in Tehran, and with Iranian insistence that the late shah, Mohammad Reza Pahlavi, had left billions of dollars in the United States.

Those two points remain among the most serious obstacles to an agreement leading to the release of the Americans. The view of Washington officials, the internal power struggle in Tehran severely limits the flexibility of Iranian authorities to modify their earlier conditons for the hostages' release. And the condition that is the most politically sensitive in both countries, as well as legally impossible for the United States to meet, is the demand for the return of the shah's wealth or for a $10 billion "guarantee" of the supposed fortune.

According to administration officials familiar with the contents of the message received from Iran through the Algerians in mid-afternoon Tuesday, there were hints of Iranian flexibility on some points -- but not on the conditions relating to the wealth of the shah.

Accompanying Christopher to Algiers last night were Assistant Secretary of State Harold H. Saunders, State Department Legal Adviser Roberts B. Owen, and Arnold L. Raphel, Iranian specialist in Muskie's office.