Labor Secretary Ray Marshall, in a move considered likely to smooth some of the ruffled feathers of organized labor, has urged President Carter to replace Betty Southard Murphy as chairman of the National Labor Relations Board.

A source close to Marshall confirmed yesterday that the secretary wrote Carter within the last two weeks suggesting John H. Fanning, a Democratic member of the board since 1957, as successor to Murphy, a Republican appointee of former President Ford.

The source said White House reaction to the proposal has been "favorable" but that no final decision has been made.

The AFL-CIO, which bitterly denounced Carter last week for endorsing a minimum wage boost that is considerably lower than the unions wanted, has not taken a position on the NLRB chairmanship. But a federation spokesman said the Fanning-for-Murphy exchange would please many union leaders who complain of pro-management bias on the part of the board, which adjudicates labor-management disputes and holds union representation elections.

Marshall's move came as something of a surprise because the NLRB is an independent agency not under control of the Labor Department. Also circumstances had appeared to rule out any quick action by Carter to change its leadership.

Members are appointed to fixed terms and only Fanning's term expires this year - in December. No Republican members' terms are due to expire until August, 1978.

However, although it has never been done before in the 41-year history of the board, a President can force a chairman to step down and elevate another board member to the chairmanship - which is what Marshall is proposing.

Murphy, warned by a friend early in the Carter administration that she might be replaced, has been fighting back. She informed some women's rights leaders who have been lobbying on her behalf, nothing that she is the only woman head of a regulatory agency (as well as the first woman to serve on the NLRB) and contending that she has done a commendable job in increasing the board's productivity and reducing delays.

"As far as I know, the President is pleased with what I'm doing," Murphy told a reporter yesterday. "We've had a 30 per cent increase in productivity without a staff increase" since her appointment, an internal reorganization and a record number of cases decided in 1976 during her first full term as chairman, she said. "I would think that's what he (Carter) wants."

Marshall was out of town and unavailable for comment but was described by sources close to him as wanting "new direction" at the top of the board to expedite decisions even more. "To some extent," the source added, "he wants someone in sync with the administration. Betty Murphy is a Republican and very upfront about it." Said another department official: "The Democrats did win the election, didn't they?"

Even if she is replaced as chairman, Murphy would be eligible to serve on the board through 1979, when her five-year term expires.

While the shift would not change the composition of the board, several union attorneys said the chairman has significant procedural powers and, according to one of them, "sets the tone" for the whole board.