Sheila Burke, a long-time top aide to Senate Majority Leader Robert J. Dole (R-Kan.), and William L. Roper, special assistant to the president for health policy, are the names being mentioned around the Health and Human Services Department as possible successors to Carolyne K. Davis as administrator of the Medicare and Medicaid programs. On Aug. 9, Davis will resign as head of the Health Care Financing Administration after serving four years.

Her departure and the expected resignation of undersecretary Charles Baker, will leave seven vacancies at the top of HHS. Some have been open for some time, in part because of disagreements between Secretary Margaret M. Heckler and the White House over who should be selected.

In addition to those two jobs, the positions of general counsel, assistant secretary for health, assistant secretary for planning and evaluation, Social Security commissioner and assistant secretary for legislation are vacant or filled only on an "acting" basis.

However, the White House is said to have approved Ron Robertson, a former Reagan-Bush campaign official, for the general counsel's job, and Food and Drug Administration Commissioner Frank E. Young to be assistant secretary for health. Sources said department officials hope to get final approval soon for the nomination of former representative Lawrence J. De Nardis (R-Conn.) as assistant secretary for legislation.

Some lower jobs are also now held by "acting" appointees: director of the Health Resources and Services Administration, director of the Indian Health Service, deputy administrator of HCFA and several HCFA associate administrator jobs.

MORE ON RHOADES CASE . . . Supporters of Dr. Everett Rhoades, the first Indian to direct the Indian Health Service, have written to Heckler, defending him in the wake of charges surrounding the circumstances under which his daughter received an Indian health scholarship to Harvard University. On June 21, Rhoades was temporarily transferred to another job pending completion of an HHS investigation.

Rhoades' daughter initially failed to receive high enough scores from two IHS graders to qualify for the scholarship. But, according to the allegations, Rhoades' aides called in a third grader who gave her a higher score, pushing her average up to the level needed to qualify.

William Byler, former executive director of the Association on American Indian Affairs, Forrest J. Gerard, former assistant secretary of Interior for Indian affairs, and Emery Johnson, who headed the IHS for 12 years, wrote Heckler on July 22, saying it was not too late "to remedy a grotesque miscarriage of justice." They contended that the HHS investigators "seriously misunderstood and misstated the facts."

The three men did not dispute that a third grader was called in, but they said that was irrelevant. Since there was money left over after scholarships were awarded to those whose scores were above the cutoff, they said the IHS lowered the cutoff level enough to use up the remaining money. They contend that under this procedure, which they said was routine, Rhoades' daughter automatically qualified with her initial score, and that the third score was not considered.

EARLY OUT? . . . Sen. John Heinz (R-Pa.), chairman of the Senate Special Committee on Aging, has charged that from July 1984 to March 1985, at least 3,700 Medicare patients were prematurely discharged from hospitals or inappropriately sent to facilities that provide less care, such as nursing homes, because of pressure on hospitals to hold costs down under the Reagan administration's new prospective payment system.

Heinz said the numbers were compiled by HHS Inspector General Richard P. Kusserow, based on information obtained by peer review organizations that monitor Medicare discharges. Heinz said the peer review groups should be given more authority to block inappropriate discharges.