Nancy Reagan's press secretary, Jennefer Hirshberg, is leaving her job after six occasionally stormy months. The White House announced yesterday that Mrs. Reagan's personal secretary and longtime assistant Elaine Crispen will replace Hirshberg, who will join the Office of Management and Budget as a communications liaison during the transition period for director-designate James C. Miller III. The changes will be effective Sept. 3.

In a statement released by the White House, Mrs. Reagan said she appreciated "the fine work of Hirshberg as press secretary and was certain her abilities would be a real asset in her new position at OMB." The same statement quoted Hirshberg as saying she valued "greatly the experience of working for Mrs. Reagan and was thankful for Mrs. Reagan's understanding and support of her career move." Hirshberg was on vacation yesterday and could not be reached.

East Wing chief-of-staff James Rosebush said Hirshberg decided to leave three weeks ago when Miller contacted her about the OMB job, and that she told Mrs. Reagan of her plans to leave two weeks ago. But one source said interviews for replacements were conducted over the past several months.

During her brief tenure in the East Wing press office, Hirshberg was involved in a series of flaps. She was seen as having inadequate access to the first lady and came under criticism from presidential West Wing staff members.

In the most recent controversy, Hirshberg's credibility came under question when she told reporters earlier this month that no biopsy would be performed on the blemish removed from President Reagan's nose. A biopsy had, in fact, been performed on the growth and a common, curable form of skin cancer was discovered.

Hirshberg's troubles began even before she moved into her White House office last February. In an interview a week before she started her job, she told The Washington Post that "the nicest thing" anybody ever said to her was that Farrah Fawcett, star of the television series "Charlie's Angels," looked like her. "I said, 'That's a twist, it's usually the other way around,' " Hirshberg recalled.

Such remarks about her resemblance to the actress reportedly displeased Mrs. Reagan, who was said to feel they made Hirshberg appear unprofessional.

Last May, Hirshberg hurried three female reporters involved in an impromptu interview with a guest at a White House lunch by pushing them and saying, "OK, come on, let's go, girls." Hirshberg later apologized to the reporters, and sources said Mrs. Reagan was upset about the incident.

While the president was hospitalized for cancer surgery, Rosebush seemed to step into Hirshberg's job as spokesman. "During the hospitalization, I did that because I was spending every day with Mrs. Reagan," Rosebush said yesterday from Rhode Island, where he was vacationing. "You'll notice that afterwards, Jennefer returned."

Hirshberg previously worked for Miller as director of the Office of Public Affairs at the Federal Trade Commission. Miller can make no staff appointments until he is confirmed, but his Senate approval is expected.

Crispen has been associated with the Reagans for years. As an employe of former White House deputy chief-of-staff Michael Deaver's public relations firm, she handled then-governor Reagan's account. Later she worked for Mrs. Reagan during the 1980 presidential campaign. As Mrs. Reagan's personal secretary, Crispen in 1982 signed a letter explaining Mrs. Reagan's decision not to accept any more donated clothing from designers. She accompanied Mrs. Reagan to the wedding of Prince Charles and Lady Diana in July 1981.

"She knows Mrs. Reagan very well," Rosebush said, "and Mrs. Reagan has great confidence in her."

Hirshberg's predecessor as Mrs. Reagan's press secretary, Sheila Tate, left the White House in February to join the public relations firm of Burson-Marsteller. She said of Crispen, "She's been with the first lady longer than anybody. She knows her very well and she has that going for her. I expect she will jump right in and be able to take over -- it's not like a brand new person."