The Arlington County School Board has become increasingly dissatisfied with School Superintendent Charles E. Nunley and some of its members have made initial inquiries about possibly removing the controversial school chief from the $58,000-a-year post he has held since 1981.

The members have sought legal advice about breaking the four-year contract with Nunley that expires in 1985, said a source close to the board who asked not to be identified.

"I'm not going to make any public negative statements about the superintendent," said School Board Chairman Evelyn Reid Syphax. "But I think it's common knowledge that there've been some problems."

When asked if she wants Nunley to serve out his contract, Syphax said, "I couldn't honestly say that."

School board member Claude Hilton said the board as a group has not discussed dismissing Nunley or buying out his contract, and other board members declined comment on the report, calling it a personnel matter. "I'm not looking to fire anybody," Hilton said yesterday.

Nunley told a reporter last night he had no indication board members are seeking to break their contract with him. "I'm not even going to address the rumor" that that might be the case, he said, adding: "Nothing has been said to me, and I talk with the school board all the time."

Last week, board members met in a closed session to evaluate the superintendent's performance. Details of that meeting are confidential, board members said, but in the past two months they have publicly expressed dissatisfaction with Nunley.

Much of the controversy has centered on what some have termed the superintendent's repeated failure to follow the board's direction on matters such as school consolidation. A final report proposing what schools should be closed by 1984 is due from Nunley in May, and one school board member said any possible action against the superintendent is unlikely before then.

Nunley's apparent troubles come at a time when the job of superintendent of a large school system is becoming more politicized and increasingly harder to hold on to. Since 1972, the average tenure of a school superintendent has dropped from 6.5 years to 5.6 years.

Nunley, a 54-year-old Ohio native, has drawn criticism from citizens, teachers and parents. They charge that he has limited their access to public school documents and generally failed to understand or encourage the traditionally high level of public participation in shaping school policies.

"It's fairly obvious that there's considerable friction," said Donn Marston, an executive board member of the Arlington County Council of PTAs and chairman of a now defunct commission that studied local school consolidation.

Nunley came to Arlington almost two years ago after the Lorain, Ohio, school board opted not to renew his contract as superintendent. Then school board chairman O.U. Johansen said at the time that Nunley was selected primarily because of his financial management skills.

Board member Margaret Bocek said Nunley has at times been a strong administrator. "I wouldn't say in the budget process, but I think he has," Bocek said. She declined to give examples.

Though largely centered around the handling of school consolidation, there have been other dissatisfactions with Nunley. He has missed board deadlines for information on consolidation and given reports board members have publicly criticized as inadequate.

Board members criticized him last fall when the superintendent prematurely named schools that could be closed. Last month, board members were critical when Nunley said information on attendance boundaries that had been requested would not be included in an upcoming report.

When the report was presented Feb. 17, board members complained that Nunley failed to include any rationale for specific school closings and directed him to do so in a follow-up report. Board members got that report last week, a few days prior to a scheduled public hearing.

Parents and citizens, meanwhile, have complained that Nunley has used the "marketability" of closed school buildings as a factor in deciding which schools should be closed.

Nunley has answered criticisms by saying that every report is not a final document and has cited cuts in staff as a reason for missing deadlines. He said considering the marketability of school buildings follows the board's wishes to lease closed buildings.