Jeane J. Kirkpatrick is looking for a new job, and President Reagan is having trouble finding one that he -- and she -- consider worthy of her talents. The lady grows impatient with her post at the United Nations, and threatens to return to private life.

Meantime, the Democratic Party is looking for a new chairman, and is having trouble finding someone the governors and the other power brokers consider of sufficient stature.

The daring, dazzling step would be to make Kirkpatrick the new Democratic national chairman. It is a match made in heaven.

You say it is outlandish to think of the U.S. ambassador to the United Nations becoming chairman of a political party? On the contrary, there is a clear historical precedent. George Bush made exactly that career change in 1973, and it became an important chapter in the r,esum,e that helped him land his present job, a heartbeat away from the presidency. That is something that might appeal to Kirkpatrick.

But, you say, Bush became chairman of his own party -- the Republican Party -- not the opposition. Ah, dear reader, you forget. Despite her starring role at last summer's Republican National Convention, the formidable Kirkpatrick is a lifelong Democrat. There would be no more apostasy in her becoming Chuck Manatt's successor at the DNC than there was in Daniel Patrick Moynihan's moving from his stint as the Ford administration ambassador to the United Nations to his present position as the Democratic senator from New York.

What are Kirkpatrick's qualifications for the post, you ask? Well, despite the demands of her diplomatic duties, it is clear that she has paid close attention to Democratic Party affairs during the past four years. The disquisition she delivered in Dallas about "the San Francisco Democrats" told them things they had never realized about themselves, like the fact that they are prone always to "blame America first." They would never have learned that from Bert Lance.

Nor was this Kirkpatrick's first demonstration of deep-delving scholarship on party affairs. In her earlier life as an academic, she published two splendid studies of the changes in political conventions and parties, one called "The Presidential Elite" and the other titled "Dismantling the Parties: Reflections on Party Reform and Party Decomposition." She is as well-equipped as anyone can be for the endless debates on party rules and delegate-selection procedures, which the Democrats employ as a narcotic to keep from thinking about the results of the last election.

As the title of her essay suggests, Kirkpatrick is not a great admirer of the steps the Democrats have taken in the last 16 years to "democratize" their party. But many other Democrats have come to think that those "reforms" have nearly run them out of business, and she would not lack for allies if she came to the chairmanship committed to leading a counterrevolution.

If you think about it, the experience that Kirkpatrick has gained in the last four years of debating Third World representatives at the United Nations is the perfect preparation for dealing with the variety of assertive caucuses that now dominate the structure of the Democratic National Committee.

If it's publicity the Democrats want, let me assure you that the reporters and cameras would be lined up outside the door, waiting for the first meeting when Kirkpatrick went toe- to-toe with the representatives of the DNC's Gay-Lesbian caucus on the issue of affirmative action.

And if the Democrats really want the wide- ranging debate on their future direction that some of them have been saying the party needs, who better to lead it than the sharp- tongued, intellectual Kirkpatrick? She was addressing -- and chastising -- her fellow-Democrats in the speech she gave the Republican convention, and if they invite her to make her points from the chairman's chair, I guarantee you there will be a vigorous debate.

Considering all the advantages Kirkpatrick possesses as a potential Democratic Party chairman, there can be only two serious questions about putting her into the post.

The first concerns the, uh, rather negative comments she made in Dallas about Walter Mondale, the titular leader and recent candidate of the Democratic Party. Her selection might be considered something of a slap in the face to Mondale -- were it not for the fact that many other Democrats are abusing other parts of his anatomy. Kirkpatrick's criticism can be faulted only on its timing; she gave it before the election.

The second question is one only the lady herself can answer: Can one who has been so lavish in her public praise of President Reagan accept the role of being the Democrats' designated hitter against Reagan and his policies? Would Kirkpatrick turn her fierce verbal artillery on the White House, whose Cabinet room she so recently adorned?

There is no way of knowing without asking her. But I do recall a saying we all learned, that "Hell hath no fury like a woman scorned."