Labor Secretary Ray Marshall yesterday asked for the resignation of Assistant Secretary Francis X. Burkhardt in response to White House pressure for a subcabinet-level shakeup within the department.

At the same time, Marshall issued a statement expressing confidence in Under Secretary Robert J. Brown, who, like Burkhardt, had been the target of an attempted purge by White House aides.

Resisting the White House pressure, Marshall said Brown will stay. "Over the last 10 days," said Marshall after discussing the matter with presidential assistant Hamilton Jordan, "I have carefully reviewed his [Brown's] performance since taking office and have come to the firm conclusion that his continued service in that office is in the best interests of the Department of Labor."

Burkhardt's departure from the post of assistant secretary for labor-management relations seemed assured, although Burkhardt said last night that he would not resign until his achievements were officially noted.

After conferring with Burkhardt, Marshall announced that Burkhardt, reached for comment by a reporter, said he would resign only after Marshall issued a statement that "fullysets forth my accomplishments" in the job.

In the statement announcing Burkhardt's impending resignation, Marshall made this assessment: "In his almost two years, he has performed a number of satisfactory and valuable services, particularly in the areas of reforming ERISA [the Employer Retirement Income Security Act program] and in establishing labor-management committees in various industries."

Burkhardt said he understood that a "more complete statement of my accomplishments" would be issued later and that he would resign after then. An aide to Marshall said such a statement will be issued sometime in the future.

Administration sources said earlier that Burkhardt and Brown were targeted, along with subcabinet officials in at least two other departments, because their work had been found by White House aides to be unsatisfactory.

However, some union sources complained that Brown, a former United Auto Workers local president before he joined the department in 1966, and Burkhardt, a painters' union official until he joined the administration last year, were the highest-ranking officials in the department who came out of the labor movement.

An AFL-CIO official called the treatment of Burkhardt a "damn outrage."

Burkhardt said yesterday that he had been faulted for being too agressive and for his "dealings" with the White House staff. He said he considered the latter citation strange because he never had any dealings with the staff. Marshall offered no reason for burkhardt's forced departure, although it was understood that he found it easier to defend Brown than Burkhardt.