Sen. Strom Thurmond (R-S.C.) shelved the nomination of the FBI's Francis M. Mullen Jr. as head of the Drug Enforcement Administration yesterday in light of complaints over the FBI's investigation last year of Secretary of Labor Raymond J. Donovan.

Thurmond, chairman of the Senate Judiciary Committee, took the action following a joint request from Senate Labor Committee Chairman Orrin G. Hatch (R-Utah) and that committee's ranking minority member, Sen. Edward M. Kennedy (D-Mass.).

Mullen, who is still one of the FBI's three executive assistant directors as well as acting DEA administrator, supervised the FBI's background investigation of Donovan's qualifications as a Cabinet appointee.

The shortcomings of that effort appear to have been compounded last week when Mullen acknowledged in an interview that he knew Donovan's name had been mentioned on an organized crime wiretap at the time of Donovan's confirmation hearings and consciously withheld that information from the Senate Labor Committee.

Mullen defended his action on grounds that the conversation, which suggested possible social ties between Donovan and a reputed member of the Genovese family of the Mafia, was "not pertinent" to the Donovan inquiry. He said he did not want to "open up" the underlying organized crime investigation and its 1,600 hours of tape recordings to Senate scrutiny.

Hatch said in an interview yesterday, however, that he cannot see how the 1979 wiretap at the Bronx, N.Y., warehouse-office of reputed Mafia soldier William P. Masselli could be so sensitive when the recordings were made available to the lawyers for Masselli and nine co-defendants following their indictments in two criminal cases last year.

"They've been given to 13 defense attorneys, as I understand it," Hatch said. "What in the world is so sensitive about them that an oversight committee of the Congress should be deprived of them?"

The tapes have never been made public. They were put back under court seal last fall when Masselli and his associates pleaded guilty without a trial to reduced charges.

Allegations that Donovan and his company had ties to organized crime and were involved in bid-rigging and union payoffs have been under investigation by special prosecutor Leon Silverman in New York since his appointment last December under the Ethics in Government Act.

In response to a request from Kennedy this week, however, Hatch pledged a formal inquiry by his committee into the handling of the January, 1981, Donovan investigation by the FBI and the incoming Reagan White House. In turn, Kennedy joined Hatch in asking that Mullen's nomination be held "in abeyance until these issues are fully resolved to the Labor committee's satisfaction."

Thurmond agreed and, spokesmen said, asked Hatch and Kennedy "to keep him informed of the status of their inquiry."

Hatch said he intends to wait until Silverman, who obtained the Masselli tapes this spring, issues his findings. Mullen has been acting DEA administrator since mid-1981. He was formally nominated for the job by President Reagan on March 29.

At a news conference Wednesday, FBI Director William H. Webster sought to defuse some of the growing dissatisfaction with the FBI's 1981 inquiry by saying that in the past FBI background checks have been designed to advise an incoming president of potential problems and not as certifications for the Senate.

Hatch and Sen. Thomas F. Eagleton (D-Mo.), another committee member, took issue with that view.

"I think they have an obligation to the people first of all," Hatch said of the FBI. "The reason for the confirmation process is that it provides some check on the executive branch by the legislative branch. . . . Certain facts which are crucial to confirmation decisions ought to be brought out." Eagleton said he was "somewhat troubled" that Webster seems to view the investigative role of the FBI as "one of fullness and frankness with the White House and something less than that for the U.S. Senate . . . . "

"I hope Mr. Webster will clarify his position in testimony before the Labor and Human Resources Committee," he said.