President Reagan, wrestling with what perhaps has become the most embarrassing action of his first year as president, took a second step yesterday toward undoing his 10-day-old decision to grant tax exemptions to private schools that discriminate racially.

The president sent a bill to Congress that would deny all such tax subsidies and ordered the Internal Revenue Service to grant only two exemptions pending congressional action on the proposed legislation.

The result leaves the administration in the position of saying its policy is to oppose tax breaks for segregated private schools, but granting Bob Jones University of South Carolina and Goldsboro Christian Schools of North Carolina tax exemptions at least until Congress acts. The federal government has charged the two schools with being racially discriminatory.

At the cost of a barrage of criticism that the president has taken a racist action, the administration has gone through the following policy changes with dizzying speed.

As of Jan. 7, the Reagan administration policy, as expressed in a brief to the Supreme Court, was the same as the policies of its three predecessors dating back to 1970: that it was a violation of the law to grant tax exemptions to schools that practice discrimination. No such schools could qualify and about 100 had been denied subsidies.

The next day, officials of the Justice and Treasury departments announced that it was wrong for the IRS to rule on exemptions in the absence of a bill passed by Congress. Suddenly, the policy had changed to grant all tax exemptions applied for by avowedly segregated schools until told not to by Congress. The officials, who declined to allow their names to be used in connection with their explanations of the policy switch, said the administration had no intention to submit legislation to Congress.

Reagan, prodded into action by the public outcry, took his first remedial step four days later. He declared that "I am unalterably opposed to racial discrimination in any form," and said that he would submit legislation to bar tax exemptions for schools that discriminate. The policy at that point had become to deplore granting such exemptions, but to allow them to be granted anyway until Congress acted.

With yesterday's action, the policy now is to deny all such exemptions except for the Bob Jones and Goldsboro Christian schools, which were in litigation before the Supreme Court when the administration made its first policy change Jan. 8, and to work for passage of a bill that would later retroactively strip the schools of their benefits.

The Reagan bill would apply retroactively to 1970 when the Nixon administration, under pressure from federal courts, stopped granting exemptions to segregated schools, many of which were founded in reaction to federal efforts to desegregate public schools.

If Congress were to pass a bill, but not make it retroactive, Bob Jones and Goldsboro would realize a windfall. Some lawyers, including Goldsboro's attorney, William G. McNairy, immediately raised the question whether making a tax bill retroactive is constitutional.

If the bill passes with the retroactive provision, administration officials said, anyone who has made a tax-deductible donation to either of these schools would then owe the Treasury additional tax money.

White House communications director David Gergen introduced two other officials to explain the latest move to reporters in the White House press room, but stipulated that they could not be identified by name.

He would not explain why officials are unwilling to stand publicly behind their latest action on tax exemptions.

The issue has triggered public criticism of each other by some of the president's top three advisers. Deputy chief of staff Michael K. Deaver has expressed concern that the initial move was taken Jan. 8 with only presidential counselor Edwin Meese III having had more than brief advance notice.

Meese told United Press International that the White House is making no "attempt to assess the blame" for the backing and filling over the tax exemptions.

"I think we're all concerned about the way it came out," Meese added. He said "I didn't have any recommendation at all," but that chief of staff James A. Baker III had taken the recommendation to the president "and I was along with him."

The right wing of the Republican Party strongly favors granting tax exemptions to the schools in question, many of which are religious schools.

The president described his bill as "sensitive to the legitimate special needs of private religious schools." It would permit schools to give preference or priority to members of a particular religious group, but would bar an exemption for any school if its "policy, program, preference or priority is based upon race or a belief that requires discrimination on the basis of race."

For Goldsboro, the Reagan bill would be a fatal blow, according to McNairy. Goldsboro would owe more than $1 million which it does not have and would have to close down, McNairy said.

Bob Jones would owe even more, but has greater resources and would not be forced to close, according to people familiar with its financial situation.