President Carter has decided to name John H. Fanning, a longtime Democratic member of the National Labor Relations Board, to succeed Republican Betty Southard Murphy as chairman of the NLRF, sources said yesterday.

The switch was recommended by Labor Secretary Ray Marshall, precipitating angry charges from Murphy that Marshall is trying to become a "labor czar" by taking control of three independent labor-related agencies outside the jurisdiction of his department.

Fanning also had the support of some labor leaders who have complained of procedural delays and pro-management bias on the part of the board, which adjudicates labor-management disputes. But organized labor was reportedly not united on the issue and the AFL-CIO took no formal stand.

Fanning, a 60-year-old attorney, has served on the board since 1957. Presumably Murphy would continue as a board member after losing the chairmanship, which she has held since she was appointed to the board by President Ford in early 1975.

Although incoming Presidents have usually chosen new chairmen, they have customarily designated outsiders for the job rather than shifting the chairmanship within the board.

Carter also has a vacancy to fill because Peter D. Walther, a Republican member of the board, submitted his resignation late last month. This will give Carter a chance to give the board a Democratic majority as well as a Democratic chairman. Previously three of the five members were Republicans.

Murphy claimed she had increased the board's productivity, but Democrats complained that she spent too much time traveling and politicking.

While in Kansas City last week, Murphy told a reporter for the Kansas City Star that Marshall was also trying to "take over" the National Mediation Board and the Federal Mediation and Conciliation Service by trying to influence Carter's appointments to these agencies. She reiterated the charges in a telephone interview from Chicago yesterday.

Donald Symthe, a Labor Department information officer, said Marshall was merely acting as the President's chief adviser on labor matters. "He's the President's chief adviser on labor and was looking at these matters in this capacity," said Smythe.

Smythe added that Marshall "felt there had been improvements [in the NLRB] and more needed to be made and that Fanning was the one to make it a more efficient operation."