After nearly a year in office, the Reagan administration finally appears to be getting around to picking a director for the Federal Mediation and Conciliation Service.

It's not that the White House hasn't tried to fill the important labor post. Late last summer the administration thought it had the job filled and was all set to send a nomination to the Senate for confirmation.

According to administration and labor sources, the White House was prepared to nominate Vincent McDonald, a former chairman of the New York State Mediation Board. The only thing left was the obligatory interview with Labor Secretary Raymond F. Donovan, which was considered little more than pro forma. McDonald never made it past the interview.

Some reports blame the interview itself for McDonald's departure. During the interview, which took place with Donovan and Undersecretary of Labor Malcolm Lovell, McDonald angered Donovan by addressing all his answers to Lovell, several sources said.

Others attribute McDonald's exit to the fact that he is a liberal Republican in the Nelson Rockefeller mold and simply not ideologically attuned to administration thinking.

In the following weeks, the White House seemed to lose interest in finding another candidate for the mediation post. Then last month the president decided to try to improve relations with organized labor--a move described by one source close to administration labor strategy as an attempt to develop a "speaking relationship" with the unions.

Apparently as part of its new effort to thaw relations with labor, the White House once again set out to find a director for the mediation service. And the new list of candidates appears impressive. According to administration sources, the White House is considering four candidates:

* D. Quinn Mills, a veteran labor relations expert and a professor at the Harvard Business School. Mills, a protege of former Labor secretary John T. Dunlop, is an economist with a background in collective bargaining in the construction industry.

* Daniel H. Kruger, a professor at Michigan State University who also is a veteran labor relations expert. Kruger reportedly has the political backing of Michigan Gov. William Milliken.

* Charles Rehmus, dean of Cornell University's School of Industrial Relations. Administration sources said they would be suprised if Rehmus--who has a strong labor-relations background, particularly in the railroad industry--would accept the post, because he only recently took the Cornell post.

* Kay McMurray, a former member of the National Medi ation Board. McMurray served on the board from 1972 to 1977.

All four men appear to be acceptable to both labor and management, but Mills seems to be a clear favorite among labor relations officials outside the government.

The search for a new mediation service director comes at a time when organized labor already has embarked on a new three-year bargaining cycle, with nearly 5 million union workers up for contract renewal in 1982.

The Teamsters already have opened negotiations for a new national freight contract with the trucking industry. The union's contract with the industry expires next March. Negotiations in the oil industry also have begun with the Oil, Chemical and Atomic Workers union amid signs there may be trouble at the table over the union's demand for a no-layoff agreement during the life of any new contract. Industry contracts expire in January.

And this week, the executive board of the United Auto Workers union voted to let the rank and file decide if it wanted to reopen the auto industry contracts early as part of an industry demand for major wage and benefit concessions. Auto industry contracts do not expire until next September.

With next year's negotiations already under way, a key mediation service official noted this week that "it would make things a lot easier if we had a director.