The State Department has approved a request by Bishop Abel Muzorewa, the first black prime minister of Zimbabwe-Rhodesia, to visit Washington early in July on a trip that could include a meeting with President Carter, according to administration sources.

The administration's decision to grant Muzorewa a visa now and the president's studied failure thus far to rule out a suggested White House meeting is certain to stir fresh controversy in Africa, where Muzorewa's white-supported government has been strongly denounced.

Nigeria and other states that have supported the administration's emphasis on the force of black nationalism in southern Africa now are warning that moves to extend recognition to Muzorewa and to end economic sanctions against the government in Salisbury could seriously affect U.S.-Africa ties.

Muzorewa is coming to Washington at the invitation of Sen. Jesse A. Helms (R-N.C.), leader of a two-year effort to end U.S. participation in the international trade embargo against Zimbabwe-Rhodesia. Helms declined to comment yesterday on Muzorewa's visit, but other sources said the Africa leader would appear before the Senate Foreign Relations Committee in an effort to bolster Helms' drive to end sanctions.

The House is due to vote today on a compromise bill that would empower the president to keep sanctions in place beyond October if he rules that lifting sanctions would damage U.S. interests abroad.

In a ruling required by 1978 legislation, Carter refused on June 7 to lift sanctions. He strongly argued then that his action protected U.S. national interests as well as meeting the legal tests established by the 1978 law, which required Carter to end the trade embargo if he found April's parliamentary elections to have been conducted freely and fairly.

But some supporters of the House bill expect the new Tory government in Britain to lift sanctions this autumn in a move that would clear the way for Carter to determine that lifting sanctions then would not harm U.S. standing abroad.

At a closed-door meeting with the Senate Foreign Relations Committee at the White House shortly before the June 7 decision was announced, Carter was asked by Helms to meet with Muzorewa, according to sources present. They said the president did not reject the idea, and administration sources say he still has not ruled it out.

Carter is said to be balancing the storm of criticism such a meeting could cause in Africa against the domestic backlash that could come from his refusal to see the newly elected black leader. Muzorewa has been cultivated by Helms and other conservatives opposed to the Africa policy enunciated by Carter and his ambassador to the United Nations, Andrew Young.

Administration officials said yesterday that Secretary of State Cyrus R. Vance, now in Tokyo with President Carter for the seven-nation economic summit, was certain to meet with Muzorewa in Washington if their schedules could be worked out. Vance met Muzorewa last year on one of the bishop's three visits to the United States.

State Department officials said it is not clear yet if Muzorewa will travel on a Rhodesian passport, which would require a special administrative clearance. They said the decision had been made to grant that clearance if needed.

A spokesman for the Zimbabwe-Rhodesia information office said Muzorewa will visit Washington "in the first half of July to discuss with interested persons and organizations the 'new reality' in his country and the progress his government is making." Other sources said Muzorewa will be in Washington July 9 and 10.

In explaining the June 7 ruling on sanctions, Carter and Vance stressed their view that elections held under the Rhodesian constitution would not be considered to be free and fair. The constitution, which was voted on only by the country's 4 percent white minority, entrenches white rule in the breakaway colony's key government departments.