"La Cage aux Folles," the Tony-winning Broadway musical comedy about the joys and heartaches of an aging homosexual couple on the Riviera, has been booked into the National Theatre for a five-month run, scheduled to begin in either mid-June or mid-September of next year.

The booking, announced yesterday by the Shubert Organization, which manages the National Theatre, follows such blockbusters as "42nd Street" and "Cats" and underscores the National's growing clout in the battle for the local entertainment dollar.

" 'Cage' has played two cities so far and it's done sensational business," said Bernard Jacobs, president of the Shubert Organization. "We think it will be as big in Washington as 'Cats,' which is the biggest show you've ever had there."

The company performing at the National will be the third to stage the Jerry Herman-Harvey Fierstein musical. The original version continues to play to capacity business at the Palace Theatre in New York after a run of more than a year. A touring company recently concluded a high-grossing engagement in San Francisco and opens later this week in Los Angeles. The Washington company will costar game show host Peter Marshall and Keene Curtis, who appeared as Daddy Warbucks in "Annie," as the temperamental gay lovers who run a popular transvestite bar in the south of France.

At a time of dwindling road show attractions, "Cage" represents a particularly potent draw for the National, which re-opened in January after a $6 million renovation and has been offering the Kennedy Center stiff box-office competition ever since. Although the National has a smaller seating capacity (1,679) than the Opera House (2,316), it can give Broadway shows the kind of long runs that are necessary to amortize the huge expenses of touring.

Roger Stevens, the Center's chairman, said yesterday that he had "had some talks" with the show's producers, who wanted "at least a 10- or 12-week run" in the Opera House. "I couldn't see tying up the Opera House for that amount of time," he said. "What they really wanted us to do was help finance the third company, which interested me not at all."

The third "Cage" company is being mounted for approximately $3 million, and theaters interested in securing the potentially lucrative booking were asked to put up $50,000 in advance for each week of playing time. The Shubert Organization would not divulge its financial participation in the tour, but independent sources say that it is in the neighborhood of $1 million. Jacobs said, however, that he did not anticipate raising ticket prices at the National, which currently has a $40 top.

Both the Kennedy Center and the National like to play down any notion of rivalry, but increasingly the big commercial Broadway hits, which used to play the Center, have been turning up at the National. After "Cats," which ends its run late in November, the Shubert Organization has booked the National with Neil Simon's autobiographical "Brighton Beach Memoirs," an all-female version of Simon's "The Odd Couple" (starring Sally Struthers and Rita Moreno) and a return engagement of "42nd Street."

"The balance of power has shifted," Jacobs said. Stevens dismissed that notion, however, by saying that the Center has some strong draws of its own in its line-up for the 1984-85 season. Among them: the Royal Shakespeare Company, which will perform a repertory of "Much Ado About Nothing" and the swashbuckling "Cyrano de Bergerac" in the Opera House for a month beginning Jan. 23; and a revival of "South Pacific," starring Richard Kiley, targeted for early spring.

The "Cage" tour will begin in December in New Orleans and play a dozen cities before reaching Washington. The date of its opening here will depend on several playing dates beforehand that have yet to be determined, Jacobs said.