Sen. Orrin G. Hatch (R-Utah) said yesterday that one of his Senate investigators has received anonymous threats warning him to "lay off" Secretary of Labor Raymond J. Donovan.

Hatch, chairman of the Senate Labor Committee, said, "These threats were serious and they were taken very seriously by members of my committee," which is investigating allegations of ties between Donovan, his New Jersey firm and organized crime.

Sources said the most explicit warning came last Friday night when the aide reported being told by a male caller that "if you keep messing with the secretary of labor," the aide, his wife and children would "end up in pine boxes."

Hatch said the matter has been referred to the FBI.

He declined to name the aide who reported receiving the two telephone warnings, but it was apparently chief committee investigator Frank Silbey.

There was no indication that Donovan was aware of the threats, the first of which reportedly came about two weeks ago.

As for himself, Hatch told reporters: "I've had some minor threats in this matter but I don't consider them significant."

He did not elaborate.

Hatch's remarks came amid what appeared to be growing indignation on Capitol Hill over the hiring of private detectives by Donovan's company, Schiavone Construction of Secaucus, N.J., to investigate the Hatch committee and its investigators.

In a letter to the Senate leadership, Hatch and the committee's ranking minority member, Edward M. Kennedy (D-Mass.), called the move "a matter of grave concern to us, transcending any partisan political concerns."

"It appears to be an assault on the institution of the Senate itself," they protested in the letter to Senate Majority Leader Howard H. Baker Jr. (R-Tenn.) and Minority Leader Robert C. Byrd (D-W.Va.).

Hatch and Kennedy asked the leaders for advice on "what formal actions are called for in order to protect the Senate against what evidently is intended as a threat to its membership, procedures and integrity."

A New York lawyer for Schiavone, where Donovan still owns a substantial interest, said the detectives were hired "on behalf of the company and its employes" and not at Donovan's request. But Hatch said Tuesday that Donovan told him a couple of months ago that "he was going to hire investigators and check on the people up here."

The labor secretary has apparently grown increasingly bitter over the committee's persistence in pursuing and occasionally publicizing allegations of ties between Donovan and his company and members of organized crime.

"We can't sit on them or hide them," Hatch said yesterday of the allegations. "The public has a right to know" about them.

"If he's innocent, then he's been treated badly," Hatch said. "But that doesn't mean we can ignore those types of things just because he's a Republican. We're not going to sit here and take much more of this stuff."

The White House, by contrast, said it had "no official curiosity" about the controversy.

Deputy White House press secretary Larry Speakes voiced the hands-off attitude in response to reporters' questions. President Reagan, Speakes said, has not "expressed a view" on the matter.

Asked if this meant the White House wasn't even curious about it, Speakes replied: "That's correct. No official curiosity. I don't think you or I know the full details of the story."

One government prosecutor told The Washington Post that the private-detective situation seems to come "perilously close" to the prohibitions of an obstruction-of-justice statute concerning congressional investigations.

When asked whether Schiavone's actions might be construed as obstruction of justice, the company's lawyer, Theodore Geiser of Newark, replied, "By no stretch of the imagination."