Striaght-arrow Republican Sen. Richard Lugar, former Eagle Scout and Rhodes Scholar once labeled "Richard Nixon's favorite mayor," spent a recent weekend in his hometown of Indianapolis, running a 3 1/2-mile foot race at a Fitness Festival, shaking hands at booths where people were getting their body fat analyzed, and delivering a Sunday sermon at a black baptist church.

None of this was unusual for him, Lugar siad, except that through it all camera crews from two television networks were dogging his steps.

For about the last two weeks, the moving finger of political handicapping has scrawled Lugar's name here and there as a possible compromise choice in the Ronald Reagan vice presidential Fitness Festival.

Accordingly, the relatively unknown Indiana senator has appeared on CBS' "Face the Nation" to discuss his views and his chances and has been chatting with a parade of reporters who are preparing profiles on him, just in case. Also, according to Senate cloakroom chit-chat, Lugar has sharpened his attire, wearing even more bankerish pinstripes than usual.

Senate Minority Leader Howard H. Baker of Tennessee is a leading contender to be Reagan's running mate, along with George Bush. But Baker would alienate the party's right wing and the betting is that the right wingers will use up some chits to get him bumped.

Lugar served as Baker's campaign chairman in the Tennessean's short-lived try for the presidential nomination -- a move which Lugar's backers see as "a bridge" to the moderates.

Unlike some of the other vice presidential hopefuls, Lugar does not have his own group or organized boosters. But he has acknowledged that he would be delighted if he is selected.

The pack journalese catchwords developed for Lugar, at this point, are "bright," but "wooden." His biggest drawback as a running mate for Reagan would be lack of both name recognition and political weight. On the other hand, it seems, he wouldn't hurt Reagan.

His pluses are his solid conservative credentials on gut issues, his performances as a pragmatic coalition and compromise-builder on key legislation, his expertise on urban problems gained as mayor of Indianapolis, and not least, the presence of his longtime political mentor, L. Keith Bulen, in the Reagan camp.

His colleagues on Capitol Hill see him as "square," and "dull," but well-informed on a number of issues and not a knee-jerk idealogue.

He played a key role in working out compromises on the New York City loan guarantee renewal and the Chrysler bailout package. In both, he insisted that the recipients of federal help agree to make greater sacrifices in return, so that the public would be, in Lugar's words, "a partner, and not a patsy."

On the other hand he is "concerned about the right [wing]" in the view of at least one Democratic senator, particularly over legislation on aid to the Sandinista government in Nicaragua.

Lugar has given the right wing an opening against him, in the view of some by favoring economic aid to Nicaragua on the grounds that the aid is "an open competition, business to business, and we should compete." (He opposed military aid because it would be government to government, directly to the leftist regime).

Right wingers led by Sen. Jesse Helms (R-N.C.), who would also like to be Reagan's running mate, ferociously opposes any aid at all. And they say they plan to raise a fuss about it.

Lugar "is just feeding the alligator [the communist regime], hoping he'll eat us last," said one conservative Capital Hill aide. He called the "open competition" in economic aid a "loaded deck," in which the benefits will be forwarded to Cuba and indirectly to the Soviets, "by secret agreement."

In any case, he said, a conservative newsletter with 400,000 subscribers will be plastered around the Detroit convention site, outlining a number of key issues including this one, on which Lugar voted with the moderate Baker and against Helms and the right.

"Still," grumped the conservative aide, "if it can't be [Helms], I like Lugar better than most of them."

Sen. Paul Tsongas, liberal Democrat of Massachusetts, worked closely with Lugar on the Chrysler bailout. Despite their philosphical differences, Tsongas described him as, "decent, very thoughful, and quite willing, if you will, to be flexible. But he is instinctively very cautious in what he does and he is not as experienced as Sen. Baker nor does he have the same grasp of the issues."

His problems with the right not-withstanding, Lugar is with them where it counts. He is strong on national defense. Indeed, in his recent CBS appearance, he indicated he would abandon the balanced budget in order to bolster the defense budget and still provide a tax cut, a position almost identical to Reagan's.

He also stood with Reagan and against Baker on the key issue of abortion rights.

"His assets dovetail beautifully with the minuses in our operation here," said Bulen, the Indianapolis attorney and political strategist who, by all accounts, put Lugar in the mayor's office in 1967. Bulen is currently in Detroit, helping organize the convention arrangements for Reagan.

Among the Reagan weaknesses and Lugar strengths he mentioned were foreign policy experience (Lugar is on the Foreign Relations Committee), agricultural knowhow (Lugar has operated a family farm), urban affairs and generally workable though admittedly "iffy" relations with the black community. g

As mayor of Indianapolis, Lugar won generally high marks, particularly for consolidating the city and surrounding Marion County into a single government called Unigov.

Nixon never actually called Lugar his favorite mayor, Lugar points out. It was a Washington Post political writer who gave him that label. He was at the time the only Republican mayor of a large city, he said, but added, "It's true there was mutual admiration."

In 1974, he unsuccessively campaigned for the Senate against Democrat Birch Bayh. He acknowledged the Nixon connection hurt him then.

But some in Indianapolis recall that he lost his own hometown and blame his failure to handle promptly a police corruption scandal there at the time. That performance, plus lingering gossip about Bulen's own questionable connections in Indianapolis, will likely get fresh scrutiny if Reagan selects Lugar.