In the slowly emerging political life of the District of Columbia under home rule, Republican mayoral candidate Arthur A. Fletcher is trying a new ploy in the campaign fund raising game: Tap the pockets of your friends and acquaintances from across the nation.

While his locally better known Democratic opponents are staging a variety of dances, luncheons and contrived events to raise money, Fletcher is attempting to collect large chunks of cash from people he worked with and met while serving as an aide to Presidents Richard Nixon and Gerald Ford.

Fletcher, a former assistant secretary of labor, developed the controversial "Philadelphia Plan" to increase the number of minorities employed in federal construction projects throughout the U.S. Partly because of that effort and other jobs he has held in the federal government, Fletcher says that some people have told him that "we look at you as the No. 1 nonelected black Republican official in the country.

The problem for Fletcher, as he readily concedes, is a simple question: "Will that admiration translate into votes?"

The task is not made easier by the fact that registered Republicans in the District are about as rare as a cool and crisp August day here. There are 182.554 registered Democrats in the city and 22.294 Republicans.

Fletcher says that within the next week or two he plans to mail a fundraising letter to between 5,000 and 10,000 people across the country, many of them long-time Republican can campaign and many of them acquaintances he has made over the years.

Fletcher already has the support of such Republican leaders as William Brock, the chairman of the Republican National Committee; George Bush, the former CIA director and former GOP national chairman; 1976 presidential contender Ronald Reagan and others.

Fletcher said that in the month since he announced his candidacy, he has raised only $5,000 (including $1,000 from Bush), but hopes that within the next month he can get enough money to show that he can be a legitimate contender. Like the major Democratic contenders, he said between $200,000 and $300,000 is needed to run an effective campaign.

He is opposed in the Sept. 12 GOP primary by Jackson R. Champion, who ran as the Republican mayoral candidate in 1974, but received only 3,501 votes out of 95,693 cast. Unlike Champion in 1974, Fletcher has the support of several of the leaders of the D.C. republican State Committee, including its chairman, Paul Hays.

While some members of the city's small Republican Party concede that Fletcher's bid is still a longshot, they said they believe that his exuberance as a campaigner and record as a public official give him a chance for an upset.

Some Republicans also said they are hoping that the spirited Democratic mayoral primary will leave the winner with a fractured party. The three major contenders in that race are Mayor Walter E. Washington, City Council Chairman Sterling Tucker and Councilman Marion Barry.

Fletcher told about 50 members of the city's Republican committee at a luncheon this week that he would like to run against the mayor, but expects he will be beaten in the Democratic primary.

"Unless he's more forceful in the next four or five weeks, something he hasn't been in the last eight or 10 years," Fletcher said of the mayor, "I seriously doubt I will be running against the incumbent."

But Fletcher said he also was "anxious to appear on any platform" with Barry and Tucker after watching them "struggle through questions and answers" at a recent campaign appearance.

"He's the best qualified candidate for mayor this year," Hays said of Fletcher. Sid Bush: "He's a leader. He's as good a public speaker there is in the United States, black or white. And he's principled.

While Fletcher is concentrating his initial fund-raising efforts outside the District, he also pleaded with the luncheon group to stage local fund raisers for him."

"I'm beginning to feel very seriously I can win it," Fletcher told the group at the Capitol Hill Club. "What it's going to take is a good organization and a whole lot of money.

"We do not have a whole lot of liquid cash," Fletcher said of himself and his wife Bernyce. "I cannot run my campaign out of my household budget. Either we get this (fund-raising effort) rolling or this thing will have to be aborted."

Noting that some business executives in Washington are giving money to the Democratic contenders as a way of showing their corporate presence in the city, Fletcher told the group, "You ought to convince them they should buy some insurance over here."

He said that after the lunch four people told him they would hold fund raisers for him.

"If we're serious about this thing," the City Council's lone Republican, Jerry A. Moore Jr., told his party colleagues, "then we've got to reach down in our pockets for some hard, cold cash."

But Moore himself said he has not made up his mind who to support in the mayoral campaign.He attended the mayor's campaign kickoff last week.

Fletcher acknowledges that some Republicans are reluctant to back him, at least partly because he has not been active in the party locally.

Others, like William M. Bartlett, the Ward 3 Republican chairman, have a different view. "He's the most articulate, knowledgeable candidate we've ever had," Bartlett said.