The night after Ronald Reagan's victory, Robert T. Thompson, a labor lawyer from Greenville, S.C., placed a phone call to Sen. Orrin G. Hatch.

It wasn't entirely a social call. Not only had a Republican been elected president but also the Senate had been placed in Republican hand, and Hatch (R-Utah) had fallen heir to the chairmanship of the Senate Labor Committee. After an exchange of pleasantries, Thompson asked Hatch if he had a candidate for secretary of labor.

He did -- Betty Southard Murphy, former chairman of the National Labor Relations Board. Thompson, who is chairman of the U.S. Chamber of Commerce labor committee, said Murphy was his candidate as well.

Such calls were standard fare in the days after Reagan's election, as leading Republicans tried to chart campaigns to put their favorites in the Reagan Cabinet. But the battle for this very sensitive Cabinet post has been marked by some unusual skirmishes, including a stormy meeting in Hatch's office in which the senator accused another labor secretary candidate -- Reagan's own transition leader -- of campaigning for the job.

Murphy has company on the list of potential secretaries of labor -- in addition to Dick Schubert, the transition adviser who felt Hatch's wrath, Rep. John N. Erlenborn (R-Ill.) is actively seeking the post and has called Vice President-elect George Bush and former president Gerald Ford.

Republicans traditionally have difficulty finding a suitable secretary of labor. In the past, Republican presidents generally selected candidates who would soothe, rather than anger, the Democrats in organized labor.

If selected, Murphy would be very much in the don't-rock-the-boat tradition of people like John Dunlop and William Usery. In ticking off her virtues, Hatch stressed that as well as being a "consummate labor lawyer, a good manager and a feminist, she is also acceptable to both sides -- labor and business."

Thompson said he called Murphy Nov. 6 to sound her out for the job. According to Thompson, Murphy "made it clear to me that she was not seeking the post, but she did give me the impression that she would take it."

Murphy does not recall the conversation with Thompson. But she did supply details of a similar talk she had with Hatch in his Senate office Nov. 6 or 7. "Hatch urged me to consider being labor secretary," Murphy said. "He told me all the reaons why he thought I'd be the best choice. I listened to him; we went back and forth."

Whether she wants the job or not, Murphy is following the established protocol of how to behave when being considered for a Cabinet post: she says she said nothing to encourage Hatch or Thompson and insists that all the efforts on her behalf are not at her behest. It is a posture of extreme public reluctance, but short of a Shermanesque statement ruling herself out of consideration.

Murphy, a partner in the law firm of Baker & Hostetler, stresses that she is "strongly motivated to stay here," but she leaves the door to the Labor Department ajar when she concedes that if the president-elect "thought I could contribute something that other people are unable to contribute, then I'd seriously consider it."

Neither Hatch nor Thompson seem discouraged by Murphy's reaction. Thompson quickly lined up a formal endorsement by the Chamber of Commerce; Hatch has called a series of Reagan insiders on her behalf.

In a widely circulated letter to Sen. Paul Laxalt (R-Nev.), a top Reagan adviser, Hatch noted that Murphy "figured in the Teamster endorsement" of Reagan. And when Reagan visited the Teamsters' Washington headquarters Nov. 18, union president Frank Fitzsimmons told the president-elect that the union wanted Murphy as secretary of labor.

Before Reagan's visit, Hatch had called both Fitzsimmons and union vice president Jackie Presser on behalf of Murphy.

Hatch has even gone so far as to accuse Schubert, a former undersecretary of labor and the Reagan transition team leader, of campaigning for the secretary's job. In a stormy meeting Nov. 25, Hatch told Schubert, "Dick, you would not be acceptable to me."

Meanwhile, Murphy's own friendships with prominent labor leaders are standing her in good stead during the transition period, just as they did during the campaign. In late August she helped arrange a meeting between Edwin Meese III, a top Reagan aide, and AFL-CIO president Lane Kirkland. In October, she took the lead in introducing Meese to prominent labor leaders at a Washington reception.

The AFL-CIO has not formally endorsed a candidate for secretary of labor, but its fondness for Murphy is clear. Thomas R. Donahue, AFL-CIO secretary-treasurer, whose friendship with Murphy began when she became the Labor Department's wage-and-hour administrator in 1974, calls her "a bright and talented administrator who understands the labor movement and the aspirations of working people."

The AFL-CIO's position could be important, because other Republican presidents have given it a de facto veto over the post of labor secretary. If the AFL-CIO were consulted by Reagan, Donahue said, officials there would approve Murphy or Schubert and reject Erlenborn.

Others close to Donahue and Kirkland take this effusive praise of Murphy with a grain of salt. "I know what Lane and Tom think of Betty Murphy," and it's not flattering, said a source close to the AFL-CIO leadership. "But she's so persistent they don't know what to do with her."

Indeed, Murphy's friends can describe her as "the world's pushiest woman" and a "skilled Washington operator" in the same sentence.

Nevertheless, a person with the active support of Hatch, the Chamber of Commerce and the Teamsters, as well as the tacit approval of the AFL-CIO, would appear to be a cinch as secretary of labor in the Reagan administration. wBut efforts on behalf of Murphy have triggered something of a counterreaction in the business community.

Thompson describes the anti-Murphy sentiments as "a little tug of war," but recent evidence suggests that may be something of an understatement.

The National Association of Manufacturers (NAM) -- the chamber's traditional rival -- endorsed five candidates for secretary of labor, including Erlenborn and Schubert. Murphy's name was pointedly not on the NAM list.

One high-ranking NAM official contends that Murphy is actively seeking the job and cites a phone call as evidence.

"I hadn't talked to Betty Murphy in over three months," he said, "when she called right after the chamber's endorsement. She began her conversation to me by saying that no matter what happens, she hoped there would be no hard feelings. She then read me the complete text of the chamber's letter."

Murphy acknowledged that she called the NAM official, but denied that she read the chamber's letter. She also denied again that she is doing anything to get the job.

All this may have little ultimate bearing on the identity of the next secretary of labor. There is just one voter in that election, and his name is Ronald Reagan.