Sen. Orrin G. Hatch (R-Utah) suggested yesterday that Labor Secretary Raymond J. Donovan should consider resigning, and warned that it would be very difficult for him to continue to function effectively.

Hatch, chairman of the Senate Labor Committee, stopped short of openly recommending Donovan's ouster, but he predicted that the beleaguered secretary's problems would get worse when special prosecutor Leon Silverman issues his findings.

"All I can say is that unless these matters can be cleared up somehow or other it is going to be very difficult for him to stay," Hatch said. "I believe that report is going to be a very difficult report."

Hatch also said the FBI gave him "additional information" Tuesday night about last year's pre-confirmation investigation of Donovan "that bothered me a great deal" because it had not been supplied to his committee in a timely fashion.

The report, it was understood, was an internal FBI memo dated Jan. 12, 1981, and dealt in new detail with alleged ties between Donovan and organized crime figures.

Hatch and the committee's ranking minority member, Sen. Edward M. Kennedy (D-Mass.), were also supplied Monday night with four other FBI documents they had not seen before, two of them described as "serious."

Hatch refused to discuss the most sensitive two, but the contents of one memo, if true, reportedly contradict Donovan's statements to the Senate committee last year about whether he knew a particular individual whose name could not be learned last night.

Special prosecutor Silverman was appointed last December under provisions of the Ethics in Government Act.

His mandate is to investigate a long train of allegations linking Donovan and his New Jersey construction company to organized crime, union corruption and bid-rigging in the New York-New Jersey area.

Silverman said yesterday that he was "working hard to complete our work at the earliest possible moment," but he denied a published account quoting him as having said his report on the matter would be submitted to the courts this week or next. "I will not predict when our task will be finished," he said.

At the White House, there was no indication that President Reagan has addressed himself to the problem. Hatch said he spoke to presidential counselor Edwin Meese III Tuesday and offered to meet with the president to brief him on the Senate Labor Committee's concerns. Meese's response was apparently noncommittal.

Donovan is now on a week-long combination work and vacation trip in Europe. Several administration sources said they felt the controversy had been so damaging to Donovan that he would have to leave, no matter what Silverman's investigation finds.

But one high official insisted, "There's no change in Donovan's status. The president won't prejudge the results of the investigation. I couldn't tell you what's going to happen. Nobody's focused on it yet."

Hatch's press secretary, Paul Smith, was quoted yesterday by United Press International as saying that Hatch "feels in the long run that he Donovan is going to have to step down."

Hatch "can see, and Donovan can see, that the handwriting is on the wall. He Hatch would prefer to wait until the special prosecutor is through, but this latest stuff has put Donovan in a sticky situation," Smith was quoted as saying.

Hatch was less direct, bobbing and weaving throughout an afternoon news conference in an effort to draw a distinction between his preferences and the pressures generated by Senate Democrats, who unanimously urged Reagan Tuesday to ask Donovan to step aside until all the allegations have been resolved satisfactorily.

"I give him the benefit of the doubt, until there is a smoking gun," Hatch said, "but there comes a time when you say, 'Can the secretary do the job?' "

The week's developments included a disclosure in The Washington Post that the incoming Reagan White House had been told by the FBI on Jan. 12, 1981, the first day of Donovan's Senate confirmation hearings, that he had "close personal and business ties with known La Cosa Nostra figures."

The FBI report also stated that "This information was corroborated by independent interviews of confidential sources." It was addressed to Fred F. Fielding, then the Reagan transition team's conflict-of-interest counsel and now White House counsel. Hatch and Kennedy said they never saw the report until last week.

A letter signed by Minority Leader Robert C. Byrd (D-W.Va.) on behalf of the 45-member Senate Democratic Policy Committee called on Reagan to ask Donovan to take a leave of absence "in the best interests of our country."

Hatch said "it will be very difficult for Secretary Donovan to continue in this position" with all the Democrats in the Senate aligned against him.

"I think the secretary of labor's got to have bipartisan support up here," Hatch said, "or at least should have bipartisan support to the extent that not all one side of the Senate is against the person."

Hatch said he expected Silverman's investigation to leave a number of allegations unresolved, enough to require follow-up hearings by his committee. Hatch already has pledged to conduct an inquiry into the FBI's performance in checking out Donovan's qualifications.

"I think there'll be a number of unresolved allegations, and we'll just have to look at them and see what we can do to resolve them," Hatch said. "If we hold hearings on this, I would think that Mr. Donovan, if he wants to continue as secretary of labor, is going to have to appear at those hearings and answer these allegations once and for all."

In the House, meanwhile, Rep. Toby Moffett (D-Conn.) introduced a resolution of inquiry seeking all records of conversations and contacts between the Reagan White House and the Justice Department "regarding the investigation of Secretary of Labor Raymond Donovan."

"The independence of the FBI, particularly when it investigates presidential nominees, must be guaranteed," Moffett said. "We need to know if interference from the White House compromised the background check system."