Rumors of Chief Justice Warren E. Burger's possible resignation this July have reached a new level despite persistent assertions by Supreme Court press officer Toni House that the scuttlebutt is groundless. The high court, with five justices over age 76, is constantly buzzing with rumors that one or another justice has decided to retire.

House derided the rumors as "silly," but they gathered momentum when it was discovered that Burger, who usually hires his clerks in early January, had selected only one clerk by early February. House insisted he had made offers to two others and was looking for a fourth.

It turns out, however, that the "two others" are current clerks who are extending for a year and that he may not be hiring a fourth clerk. Burger will not say which of his current clerks are staying on. Justices, including Burger, occasionally have kept clerks on for a second tour. But it is highly unusual to keep two on for a second year; a knowledgeable source could not recall Burger's ever having done so.

One source said Burger, who is spending increasing amounts of time chairing the Constitution bicentennial commission (some suggest that this new passion of Burger's might explain a decision to resign), simply felt it would be easier to keep some of his current clerks than to break in new ones.

Another source argued that Burger is suffering from acute "Gramm-Rudman-itis" and feels it may be appropriate to make do with three clerks next term.

All sources indicated that if Burger is indeed contemplating a voluntary departure, it most likely would be on Sept. 17, 1987 -- the 200th anniversary of the Constitution's signing and Burger's 80th birthday.

Burger himself is giving no indication that he is thinking of leaving.

Trial by Festival . . . The Smithsonian Institution's 20th Annual Festival of American Folklife will feature, according to a news release, "Japan, Tennessee, Trial Lawyers and Traditional Crafts."

The release explains that lawyers qualify for this celebration of "this nation's rich cultural diversity and the many international traditions from which it springs" by virtue of "the occupational folk traditions of trial lawyers." The program is funded by a two-year grant from the Association of Trial Lawyers of America.

Washington has long been recognized as a fine place to watch lawyers in their natural habitats, but the coming festival, which straddles the July 4 weekend, may be the first opportunity for Washingtonians to watch them on the Mall.

On Another Note . . . Sen. William Proxmire (D-Wis.) has asked the General Accounting Office to investigate the Supreme Court's use of its motor vehicle pool.

Proxmire, in a letter March 6 to Comptroller General Charles A. Bowsher, cited a Washington Post article last month reporting use of the court's cars for personal missions. Proxmire said the alleged uses "appear to be inconsistent with your guidelines regarding the appropriate use of federal vehicles under current law." He asked for a response by late May.

Lone Star Statement . . . Last Friday, Education Secretary William J. Bennett took questions from editors and publishers at the National Newspaper Association's annual convention. Asked by a Texas editor about his support for the state's controversial new "No pass, no play" policy for high school athletes, he said, "It's tremendous . . . . If Texas can come out and say English is more important than football, anybody can say it."