The White House said yesterday it has begun its own inquiry into the case of Bill Milliken, a close Carter family friend who interceded with the government for a developer who agreed to give $10,000 to a nonprofit corporation Milliken runs.

Presidential press secretary Jody Powell said the investigation, which may be completed today, will seek to determine whether Milliken's actions were improper or gave the appearance of impropriety, and what actions, if any should be taken.

Powell sidestepped most questions involving the case, pending the outcome of the inqury, which is being handled by White House counsel Robert Lipshultz.

Milliken conceded ysterday that "I make a mistake in judgment . . . I'm just not used to this area," meaning the world of Washington politics. He said it "looks bad," but insisted, "It was all innocent."

United Press International quoted Milliken as saying that this was the only time he contacted anyone in the federal government on behalf of an outsider.

Rosalynn Carter, Milliken's strongest supporter in the White House, had no comment on the case, according to Mary Hoyt, the First Lady's press secretary.

"I'm sure she will be talking to him," Hoyt added.

She said Milliken sometimes visits Mrs. Carter in her office to discuss urban problems. Milliken, who lives in Atlanta, he said that when in Washington he sometimes stays overnight at the White House.

"Bill has been a friend," Hoyt said. "He suggested the prayer at the opening" of the Camp David summit conference - a reference to an appeal for world prayer issued on the first day of the summit by President Carter. Egyptian President Anwar Sadat and Israeli Prime Minister Menachem Begin.

The Washington Post reported yesterday that, earlier this year, Milliken interceded with federal housing officials on behalf of George Zamias, a wealthy Johnstown, Pa., developer seeking a $15.5 million federal grant for a shopping mall-apartment-hotel complex.

While discussing Zamias' efforts to obtain a federal grant, both men have acknowledged in interviews, Milliken solicited and later received a $10,000 contribution from the developer as a contribution to some of his corporations.

Milliken later called a friend in the administration, Richard Fleming, deputy assistant secretary of housing and urban development, who set up a meeting between Zamias and HUD officials.

Zamias' grant application is still pending.

Milliken is widely known as a deeply religious man who heads several non-profit corporations that provide assistance to inner-city children experiencing trouble in school and to street academies for dropouts and youths with personal problems.

His interest in these ventures attracted the attention of Rosalynn Carter, who press secretary has described Milliken as an unpaid adviser to the First Lady.

In addition, although he does not work for the government, Miliken has secured $1.2 million in federal funds for his projects and office spare for two of the ventures in the Old Executive Office Building, which houses most of the White House staff.

Powell said the White House investigation began Wednesday after reporters for The Post raised questions about Milliken's role in interceding with HUD. He said the president was informed of the inquiry yesterday and agreed it should be conducted.

Powell said Lipshutz's office was rushing to complete the inqury as soon as possible.

"The goal is to see it remedial or preventive action is needed," he said.