Within the high councils of the White House, there is heightened expectation that Office of Management and Budget Director David A. Stockman will voluntarily leave the administration early next year after the third Reagan administration budget is submitted to Congress.

"He's lost his stroke with the president," one administration official said of Stockman, who also has attracted many critics among influential Republican senators and House members. The view that Stockman will leave, possibly in February, is fueled by the fact that some of his ablest associates are negotiating for jobs outside the administration.

Stockman insists that these reports are either rumors or wishful thinking. "I am absolutely not leaving," he told one of his closest aides last week.

Nonetheless, there is much talk at the White House that the president will be shopping late this winter for an OMB replacement. Presidential assistant Richard Darman, a friend of Stockman, has been mentioned frequently as a successor, but Darman would face strong opposition from the ultra-conservative camp. Another possibility is Transportation Secretary Drew Lewis, who already has turned down an offer to become Republican Party chairman.

The OMB job is more important than the party chairmanship but equally thankless. Preliminary estimates within the administration indicate a probable budget deficit in excess of $160 billion in the 1984 fiscal year.

If the president takes a Far Eastern trip in 1983, a possibility in late winter or early spring, Japan will be on his itinerary, administration officials have disclosed. But they say this trip, which would include a visit to Korea, is still in the discussion stages.

The Korean foreign ministry announced in Seoul last week that Reagan would visit Korea. A senior administration official said subsequently that no promise has been made but confirmed that Korea has issued an unofficial invitation and that U.S. officials have told Korea that the president would like to make the trip sometime.

The Far Eastern trip, if it comes, definitely will not include a journey to China, these officials say. They believe Chinese leaders should come to Washington first.

Around the White House, Interior Secretary James G. Watt has been nicknamed "E.T." after the lovable extraterrestrial in the movie of the same name. But high-ranking administration officials aren't amused by Watt's determination to have himself photographed with antelope he has gunned down. These officials are even less amused by his proposal to allow the killing of mountain lions in some western national parks on the grounds that they are eating sheep.

Watt's in-house critics say that administration electoral prospects are already difficult enough in the western part of the country and that it's bad politics to be inciting environmentalists at a time like this.

At a White House gala one week from tomorrow, Nancy Reagan will unveil her new book, "To Love A Child." Profits go to the foster grandparent program with which the First Lady has long been associated. For the occasion, Frank Sinatra will sing a new song with the same title as the book. . . The White House is planning a state dinner in San Francisco for Queen Elizabeth II when she comes to the United States for an extended visit in March.

Don't look for President Reagan at Wednesday night's tribute to Labor Secretary Raymond J. Donovan in Washington. Even before Reagan scheduled a nationally televised speech the same evening centered around his signing of the jobs bill, sponsors of the Donovan dinner, being given by friends of the secretary, had gently been informed that Reagan wasn't coming. One of the principal speakers, however, will be White House counselor Edwin Meese III.

It's not true that the dinner will be expanded into an all-purpose salute to Donovan, departing Republican chairman Richard Richards and veterans administrator Robert Nimmo, who stepped down last week just ahead of a General Accounting Office report expected to be severely critical of his conduct in office.

White House political strategists have grown pessimistic about the chances of Chic Hecht, GOP challenger to Democratic Sen. Howard W. Cannon in Nevada. Cannon has proved a much stronger candidate than anticipated and Hecht a weaker one. On the other hand, the White House has grown decidedly bullish about the prospects that Republican Larry Williams can upset Democratic Sen. John Melcher in Montana. Reagan, who has made one trip to Montana for Williams, is likely to make another in the last two weeks of the campaign.

Reaganism Of The Week, from a speech to veterans organizations in Columbus, Ohio: "So what is really back of our great attempt to refurbish our military is to strengthen three delegations of Americans, two of which are in Geneva, and one, I believe, still in Switzerland."