The Fairfax County school administration proposed a sharp increase in school spending yesterday that would increase real estate taxes on the average Fairfax county home $113 a year and make the county's new teachers the second-highest paid in the area.

The proposed budget calls for the county to spend 15 percent, or $42.4 million, more on its schools during the coming fiscal year than it currently does. Most of the increase would be used to grant the system's 12,400 workers an across-the-board 10.4 percent pay increase.

Overall, the budget, prepared by acting school superintendent William J. Burkholder, calls for Fairfax to spend $327 million for its school system, the largest in the area.

If approved by the County School Board and Board of Supervisors, the pay increases would increase starting teacher pay in the county to $12,675 a year, second in the region to the District of Columbia's starting level of $13,347. Beginning teachers in the county currently are paid $11,041 a year.

County politicians who must act on the request were lukewarm yesterday.

Fairfax Board Chairman John F. Herrity, who has promised to keep county taxes low, sounded dubious. "I probably could slam the thing," Herrity remarked yesterday. "But I think it's premature until I know what's in it and until we get further along in the budget process."

Blaming some of the increased costs on the skyrocketing energy costs, Burkholder said the proposed budget would allow the school system to regain ground lost to inflation since the mid 1970s.

"This budget was developed to be realistic and responsive to the needs of the school system in order for us to continue quality education," Burkholder said yesterday. "In past years, certainly in recent years, economic and cost restraints have held our school budget to austere fiscal plans . . . [and that] has resulted in programs . . . that have been below the minimum . . . level . . . necessary for quality instruction."

Although the budget was prepared by acting superintendent Burkholder and his staff, it will be Linton Deck, the newly appointed school superintendent who takes office later this month, who must win approval of the spending plan.

Deck, currently an Orlando, Fla., superintendent, said yesterday he was "as comfortable [with the budget] as you can be in the position I'm in."

Burkholder said yesterday he hoped the proposed salary increase would alleviate teacher frustration in the county. "I don't think its any secret that employe morale, particularly teacher morale, has been at a low level during the past couple of years," he said.

Fairfax county teachers have been engaged in a "work-to-the-rule" job action since last spring when members of the Fairfax Education Association voted to refuse extra, unpaid out-of-classroom work to protest low pay increases.

Fairfax Education Association president Gerry Gripper has said that he was pleased with the proposed increase, and indicated that its inclusion in the superintendent's budget might be enough to end the job action. Members of the teacher's group are expected to vote Jan. 16 on whether to continue the protest.

Burkholder said that energy costs have risen so high that the county may have to undertake "less traditional ways of dealing with this problem." He said the county may ultimately want to consider closing schools during the coldest months of the year, going to four-day weeks during the winter and cutting school bus routes.

"Recognizing that school buses alone travel 9 billion miles in Fairfax County, you can appreciate the kind of impact that has on our budget. Add to that fuel oil costs . . . [and] this budget has $16.2 million to cover energy costs. That may very well not be enough," he said.

"The key thing in this budget may very well be state aid -- $10.5 million worth of state aid," said board chairman Rodney Page. "In fact, the biggest obstacle -- may not be the General Assembly but the governor who is asking for large budget reductions."

If a state aid is cut, school administrators say they will seek those funds from the county government.

Fairfax officials emphasized that although the dollar amount of the county contribution has increased $27 million, the percentage of the local money in the total budget will remain the same -- 64.1 percent -- in 1981 as it is in the current year if state school spending is not cut.

Among the expenditures to be covered under the proposed budget are:

Hiring 325 new employes.

Opening of four new schools.

Reducing class size for grades one through six.

Replacing 70 school buses.

In response to declining student enrollments in the eastern section of Fairfax County, school administrators embarked last month on a study to determine which, if any, schools in that region would be closed at the end of the year. Officials said yesterday that no school closings were taken into consideration in the budget.

A public hearing on the superintendent's proposed budget is scheduled for Jan. 28 at Falls Church High School.