President Pieter W. Botha announced today that he had agreed to meet with Nobel laureate Bishop Desmond Tutu to discuss South Africa's racial crisis. The timing of the meeting was uncertain, but there were indications that it could take place next week.

Tutu first raised the possibility of a meeting yesterday. Botha, after an initially chilly response, said later that he was open to talks, and Tutu sent his formal request today.

The president's office announced Botha's willingness to talk with Tutu as the number of arrests under the week-old state of emergency rose to 1,086 today, and as South Africa braced itself for intensified international pressures following last night's U.N. Security Council vote for voluntary sanctions in support of demands that the Botha government lift the emergency.

Foreign Minister Roelof F. (Pik) Botha warned South Africans at a meeting in the northern Transvaal mining town of Phalaborwa tonight that they should prepare to withstand a "concerted economic and diplomatic onslaught" against their country.

Violence continued today with rioting in the black township of Guguletu, near Cape Town, and a black youth was seriously injured while trying to throw a gasoline bomb at a black policeman's home near Port Elizabeth. But police reports indicated a relatively quiet day countrywide, suggesting that the mass arrests of black community leaders may be beginning to quell the unrest.

The situation in the segregated black townships remains tense, however, and Botha's willingness to meet with Tutu is being welcomed in some quarters as a conciliatory gesture, although few people expect the meeting to yield any substantive agreement.

Tutu, who is highly regarded by the radicals involved in the wave of rebellion in the black ghettos, offered to mediate with the government in the course of a British Broadcasting Corp. interview yesterday. He elaborated on his offer in another interview later.

"I would be prepared to talk to the president, provided it was not something that he was doing to have a kind of talk shop," Tutu said.

Such a meeting would have to focus on the unrest and how to end it. "We would have to get down to brass tacks pretty quickly," Tutu added.

The president's immediate reaction was chilly. He said that he could not respond to offers made through the media but added that he was prepared to talk to anyone who did not propagate violence, "provided a proper appointment is made."

Tutu immediately sent a formal request for a meeting, and today President Botha's secretary, Jack Viviers, announced in Pretoria that a meeting would be arranged "at the state president's earliest convenience."

Viviers said the president had also received a request for a meeting from the archbishop of Cape Town, Philip Russell, who is head of the South African branch of the Anglican Church to which Tutu belongs.

"We will have to see when Bishop Tutu can be fitted into the president's busy schedule," Viviers added.

The meeting involves political risks for both men. Despite his international reputation and a recent act of courage in preventing an angry black mob from killing a suspected police informer, Tutu is still regarded as a dangerous radical by most white South Africans.

Botha could find himself accused of appeasement at a time when his followers are demanding tough action to crush black unrest.

The fact that Botha is prepared to risk such criticism is an indication of his government's concern at the deepening internal and international crisis it is facing as a result of the 10 months of racial unrest and the state of emergency it declared July 20.

For his part, Tutu risks losing credibility among the black radicals by seeming to negotiate with the government. Total rejection of the white administration, and particularly of blacks who collaborate or negotiate with it, has been the theme of their rebellion.

Already Tutu has lost some standing in the eyes of the young radicals by appealing for an end to their campaign of vengeance against suspected police informers and other blacks regarded as "collaborators" with the government.

Botha and Tutu met once before, in 1980, in a meeting that yielded little. Tutu spoke about the need to dismantle the segregationist system called apartheid as an overture to a nonracial political settlement, while Botha dwelled on what he called the terrorist threat to South Africa.