Former Conservative prime minister Edward Heath has become an embarrasing one-man employment problem for Britian's new Conservative prime minister, Margaret Thatcher.

This week Heath turned down Thatcher's offer to make him British ambassador to the United States.

Thatcher, who deposed Heath as Conservative Party leader in 1975 and did not include him in her Cabinet after she became prime minister two weeks ago, then sent word that she would be willing to nominate Heath for secretary general of NATO. Heath turned that down, too.

Thatcher's inability to find a proper place for Heath both conflicts with her reputation for tidiness and decisive action and leaves a nagging political problem unsettled.

Heath remains a popular, important figure in the liberal wing of the Conservative Party, from whom Thatcher and her rightist followers seized power to begin their crusade against socialism. His brooding presence on the Conservative Party's front beach in the House of Commons - just across an aisle from Thatcher and her Cabinet ministers - is seen by some Thatcher loyalists as a threat.

Heath's personal dislike for Thatcher and strong disagreement with some of her policies is well known, although never stated in public oratory. While delivering vigorous, effective speeches for the Conservatives during the recent election campaign, Heath never mentioned Thatcher, referring only to "the leadership" or "they".

Former Labor prime minister James Calagham mischevously commented during the campaign that Heath "regards himself as the rightful leader of the Conservative Party in exile. She stabbed him in the back when he was a leader, and he has never forgiven her."

After the election, Thatcher balanced the right-wingers she put in her Cabinet with a number of "Heath men" who had worked closely with him in the last Conservative government in the early 1970s. Yet Heath himself was left out, which dominated the headlines in the popular press here.

Heath's supporters had touted the silver-haired 62-years-old for foreign secretary. He had taken Britain into the Common Market and had maintained his contacts with world leaders after losing the party leadership. He continued to travel widely and share his love of good food and wine at home with a wide range of friendly diplomats.

Nevertheless, Thatcher, who never spoke to Heath about it, sent him a letter saying that she had chosen Lord Peter Carrington as foreign minister. That filled the one post that seemed suitable and sufficiently important for Heath in her government.

Then this Monday, another note from Thatcher was delivered by a government messenger to Heath's London townhouse. In it, Thatcher told Heath that she thought he would do a "superb job" as ambassador to Washington, the post now held by former prime minister Callaghan's son-in-law, Peter Jay.

Jay routinely submitted his resignation when the Conservative government took office. Although it has not yet been accepted, he is to be replaced. Reports circulating in diplomatic circles here say that Jay, a former journalist and television commentator, plans to take a foundation or university position and stay in the United States.

Heath, however, answered Thatcher in a note of his own on Monday.

"Dear Margaret," it reportedly read, "thank your for your note. As I have said, I do wish to stay in the commons. I am sure you will be able to find somebody to do the job well. Yours, Ted."

Then, according to informed sources, Thathcer asked Carrington to have senior Foreign Office friends of Heath sound him out on the NATO secretary general's post. Although it is not her job to offer, Thatcher could propose Heath for it and he likely would be a polular choice to succeed the present secretary general, Joseph Luns.

Heath sent back word that he was not interested in that job either.

Heath, who was in his customary seat for an important foreign policy debate in the House of Commons today, would not discuss the job offers with reporters.Thatcher's office said it would not discuss the prime minister's private correspondence.

With the Thatcher government otherwise settling in comfortably and the murder conspiracy trial of former Liberal Party leader Jeremy Thorpe ploding through most of the same testimony that became public at hearings last year, the Heath question remains Topic A for political gossip here.

Even a journal for the social sciences, New Society magazine, joined in with an adroit parody of A.A. Milne's "Winnie-the-Pooh," featuring Thatcher as Christopher Robin, Heath as Heffalump the elephant, and Thatcher's top Cabinet members and advisers as Winnie-the-Pooh, Piglet, Eeoyre and the other animals of Hundred Acre Wood, each appropriately cast for his own idiosyncracies.

"It's time we decided what to do about the Heffalump," Christopher Robin tells the others at the beginning of the tale. "We can't just let him go on heffalumping around Hundred Acre Wood."

In the end, however, they can't decide just what to do about him, so they decide just to send him a letter. CAPTION: Picture 1, EDWARD HEATH, Financial Times; Picture 2, MARGARET THATCHER, UPI