Several months after the Fairfax County School Board offered Superintendent William J. Burkholder a new $144,500 a year contract, it privately urged him to fill a $65,000 to $70,000 job in his administration.

Burkholder said last night that he plans to ask the board to fill the position of a second deputy superintendent that he has left vacant since he took office more than two years ago.

The job will be advertised by the School Board this week, according to Warren Eisenhower, assistant superintendent for personnel services.

School Board members said yesterday that they suggested last spring Burkholder add a second deputy to his administration to guarantee continuity at the top level of the system. Both Burkholder and Donald Lacey, now the single deputy superintendent, were expected to retire next July.

"We came to him and asked him to do this," said at-large School Board member Joy G. Korologos. "We felt the position should be restored to give continuity to the school system at the top."

After agreeing two weeks ago to remain under a new four-year contract, Burkholder announced last Thursday he would resign effective next July. His decision followed a public outcry over the $157,000-a-year pay package he had been promised by the School Board for the next four years. Lacey could not be reached yesterday to say whether he intends to leave his job in July.

Shortly after Burkholder took over as superintendent in 1982, he announced he would appoint only one deputy, saying one was all he needed.

The new deputy will relieve what School Board members said is the excessive workload of Burkholder's top assistants. One deputy will be responsible for instructional matters and the other for personnel and business matters, board members said.

Among those mentioned for the deputy superintendent's job are Area Four Superintendent Arthur Gosling and Area Three Superintendent Herman A. Howard. Neither one would say yesterday whether he intends to apply for the position. Burkholder will make a recommendation to the board on who should get the job, which pays up to $70,000.

Howard, who is black, was at the center of a controversy two years ago over Burkholder's decision to eliminate one of the deputy posts. In September 1982, after less than three months on the job, Burkholder came under fire from representatives of several black groups for eliminating the job.