Washington — Former President Jimmy Carter tells a humorous and obviously exaggerated story about how he selected his vice-presidential running mate. He asked several people to fly down to Plains, Ga., over a period of several days for quiet talks. None of them knew much about the South, but each of them tried to make a good impression.
One eager politician stepped off the plane at Plains and exuded: ''Jimmy, I just can't wait to see the peanut trees!'' Others made equally wild comments. So by the time Walter Mondale arrived for his interview, Mr. Carter was desperate. As the door opened on Mr. Mondale's plane, Carter rushed up to him and said: ''Fritz, just don't say anything, and the job is yours.''
Now it's Walter Mondale's turn to do the picking. This time, however, the atmosphere is a little different from when Carter made his choice in 1976. There are still many potential vice-presidents. But unlike 1976, this time there is one name - Gary Hart - at the top of just about everyone's list. This city now wonders:
Will Mondale offer the No. 2 spot to Senator Hart, despite their often-bitter primary battles?
And will Hart accept the offer, if it comes?
Former Democratic contender George McGovern, who has been talking privately with both Mondale and Hart, wants to see a Mondale-Hart ticket. But he says Mondale is not rushing into any decision. The former vice-president has asked Washington lawyer John R. Reilly, a senior adviser, to put together a list of potential running mates, and to begin making background checks on them.
Meanwhile, Mr. McGovern says, Mondale himself is giving careful thought to all sorts of factors. For instance, would a Southerner like Florida Gov. Robert Graham help the Democratic ticket this fall? Or a Texan, like Gov. Mark White? Or a Westerner, like Senator Hart? Or a woman, like Rep. Geraldine A. Ferraro of New York? What about a black, such as Mayor Tom Bradley of Los Angeles, or a Hispanic-American, such as Mayor Henry G. Cisneros of San Antonio? What about combining several qualities, such as a woman who comes from the West, and is also a Hart ally, like Rep. Patricia Schroeder of Colorado?
Every discussion here, however, seems to come back to Senator Hart.
Hart seems cool to the idea of teaming up with Mondale. The senator has fought long and hard. He still has not given up hope of heading the ticket. Something unexpected could still happen to throw the convention his way, such as a legal decision against certain Mondale delegates, he suggests.
Behind the scenes, however, Hart's own friends are urging him to think seriously about the vice-presidency. He appears to be listening to them with more and more interest, they say.
They argue, first of all, that Mondale may surprise the experts and win this election against President Reagan. If he does, then Hart, as the vice-president, would be in an advantageous position, the heir apparent who would be first in line to run following President Mondale's term.
Of course, Mondale might lose. Even so, Hart's friends say, the senator would be helped by taking the No. 2 spot on a Mondale ticket.
In 1988, a horde of Democrats are expected to run for the White House, if Reagan wins this year. In the Senate there are more than a half-dozen, including Edward M. Kennedy, John Glenn, Dale Bumpers, Sam Nunn, Joseph R. Biden Jr., Bill Bradley, and probably others. Outside Washington there are some well-known names , such as John D. Rockefeller IV, the governor of West Virginia, and Mario M. Cuomo, the governor of New York. The Rev. Jesse Jackson will probably be back. And it would surprise no one to see a woman or two making a serious try.
In this kind of crowd, Hart will need all the help he can get. If he runs for vice-president this year, his friends are telling him, that will make him the presumptive leader of the party for the next four years, and strengthen his hand immeasurably in 1988.
The immediate concern for most Democrats is not 1988, of course. It is 1984. And this is what makes Hart look attractive to many Democrats for No. 2.
A Gallup poll released this week indicates that Hart does seem to provide a boost to Mondale's chances against Reagan. In a Reagan-Bush vs. Mondale-Hart showdown, the Reagan team edges by with only a four-point margin, 51 to 47, according to the poll. Without Hart's help, pitting simply Reagan vs. Mondale, the President leads by nine points, 53 to 44.
This five-point boost given by Hart could be crucial. Since 1960, 3 out of 6 presidential elections have been decided by less than two points. Two of those were settled by less than a point.
Mondale-Hart would also help to heal the party. This year's campaign has uncovered deep fissures among Democrats - young vs. old, rich vs. poor, white-collar vs. blue-collar, union vs. nonunion, East vs. West, city vs. suburb. Mondale needs help to bridge these gaps, and Hart is one obvious choice to do it.
In this quiet period before the convention, there are indications that both Mondale and Hart are seriously exploring a possible linkup. Top aides from both sides, including Hart campaign manager Oliver Henkel Jr. and Mondale campaign manager Robert Beckel, have been in contact with one another. Mondale has shown himself willing to be the first to pick up a phone to begin talks with both Hart and Jackson to begin the healing process.
One final point. Privately, Hart's friends say he feels strongly about certain issues. He does not feel willing to compromise, for example, on his opposition to trade protectionism, as favored by Mondale for automakers.
If Hart runs with Mondale, one friend says, the senator might insist that he and Mondale ''agree to disagree'' on certain fundamental issues. Whether Mondale could swallow such a deal is uncertain.