The Reagan administration yesterday asked the Supreme Court to settle the politically explosive issue of whether segregated private schools are entitled to tax exemptions. But it took the unusual step of suggesting that the court select another party to defend the Internal Revenue Service's authority to deny such exemptions.

The administration created a controversy Jan. 8 by announcing that it was reversing the longstanding IRS policy, and administration officials emphasized yesterday that their decision to pursue the case does not change their opinion that the IRS has barred the exemptions illegally since 1970.

The government said in January that a pending case involving Bob Jones University in Greenville, S.C., and Goldsboro, N.C., Christian School should be dismissed because the IRS planned to grant their disputed exemptions. An appeals court blocked that promise last week, and government attorneys said yesterday, "Our previous suggestion of mootness is therefore no longer appropriate."

Because the government now says the case should be heard but backs the schools' position that the law permits the exemptions, the administration suggested that the Supreme Court find another party to argue the opposite view.

Lawyers for the NAACP and the Lawyers Committee for Civil Rights, who also have suggested that a third party argue the original IRS position, said they are delighted with the latest government switch in the case. Congressional leaders, who showed no enthusiasm for pushing an administration bill specifically banning such exemptions, agreed.

William Bentley Ball, an attorney for Bob Jones University, said he opposes having someone else argue that the IRS policy is legal. Even before the government's motions yesterday, Ball filed a motion opposing the "divided argument" idea as unfair. He said the court should not be asked to manufacture a controversy by picking a party not previously involved.

In a telephone interview later, Ball said that if one "friend of the court" is allowed to argue for racial civil rights another should be allowed to argue for religious civil rights.

Thomas Atkins, general counsel for the NAACP, said, "I think the government has made the best out of a bad situation. They seemed to recognize that Congress wasn't going to bail them out, so the case should go forward."

Norman Chachkin, of the Lawyers Committee, said, "It's good to see the case back where it belongs."

Rep. Dan Rostenkowski (D-Ill.), chairman of the House Ways and Means Committee, said in a statement, "The administration has finally acknowledged that the Supreme Court is the proper forum to judge this issue."

Sen. Robert J. Dole (R-Kan.), chairman of the Senate Finance Committee, which also held a hearing on the administration bill, has said he hoped the Supreme Court, rather than Congress, could settle the issue.

Some lawyers expect the Supreme Court to consider the case at its conference today, but a decision probably would not be released until Monday at the earliest.

Bob Jones University and the Goldsboro school have racially discriminatory policies based on what they claim are sincere religious beliefs. Bob Jones bars interracial dating and marriage, while Goldsboro's admissions policy bars blacks.

White House communications director David R. Gergen went to some lengths to say the latest twist in the Bob Jones case is a Justice Department decision. "This is not a reversal. It's a reaction to the court of appeals decision," Gergen said.

Last September Justice attorneys filed a brief at the Supreme Court defending the IRS policy of denying exemptions to the two schools. The controversy began when the government went back to court Jan. 8 and said the IRS was wrong because the tax exemption statute did not refer specifically to racial discrimination as a bar.

In the court papers filed yesterday acting Solicitor General Lawrence G. Wallace said that while Justice supported the schools on the statutory argument the government brief would oppose them on their First Amendment religious claim.

The government thus suggested that the usual one-hour oral argument in the case be divided into thirds among the schools, the government and the outside party picked by the court to support the original IRS position.