The government's new Institute for Museum Services should have a board, and a director, by the middle of next month, a White House spokesman said yesterday. Should Senate approval be forthcoming, their names will be announced at a meeting scheduled tentatively for mid-December at the Brooklyn Children's Museum in New York.

The Institute, a branch of the Department of Health, Education and Welfare, will this fiscal year distribute some $4 million to American museums of history, art and science. Though government agencies, among them the National Endowments for the Arts and the Humanities, already contribute funds for museum exhibitions, the new Institute is authorized to help defray operating expenses.

President Carter is expected to nominate Lee Kimche, 43, of Washington, a former official of both the American Association of Museums and the Association of Science-Technology Centers, to the $47,500 post as the Institute's director. An FBI background investigation is now under way, Kimche has been working as a consultant at HEW since August.

The names of 15 board members were sent to the Senate last month by the President. Those nominated include George Seybolt, president of the Wm. Underwood Co., Boston, and director emeritus of the Boston Museum of Fine Arts, who has been nominated board chairman: Joan Mondale, the wife of the Vice President; C. Douglas Dillon, president of the Metropolitan Museum of Art; Lloyd Hezekiah, director of the Brooklyn Children's Museum; Peter H. Raven, director of the Missouri Botanical Garden and architect Lewis Davis of New York.

Also Raul A. Lopez of UCLA's Museum of Cultural History; E. Leland Webber, director of the Field Museum of Natural History, Chicago; and Gary K. Clarke, director of the Topeka Zoological Park. A congressional aide said yesterday that a lag in the submission of supplementary documents on those nominated has delayed Senate action.

Though the President has not yet nominated Kimche, Seybolt said yesterday, "As far as I know, no one else is even being considered for the director's job."