The Falls Church school administration has proposed cutting back its teaching staff and elementary school classes and dropping its swimming program to combat escalating school costs and federal budget reductions.

Despite the recommended cuts, the administration has asked the school board for a 5.5 percent overall increase in the budget for Northern Virginia's smallest public school system. The 1982-83 proposed budget totals $5.2 million, up $273,000 from this year's school budget.

"It's going to be a pretty tight year," said School Superintendent Warren J. Pace. "With the federal cutbacks and inflation, we're going to have to come up with more local money."

Pace said he is prepared for a tough fight with city officials, who say they are facing declining city tax revenues. The city's share of the budget would jump 7.3 percent under Pace's proposals because of reductions in federal contributions.

The superintendent said he will recommend cutting full-time kindergarten classes to half a day and eliminating bus transportation of students at George Mason Junior-Senior High School if school board and City Council members don't meet the budget requests.

The Falls Church School Board will vote on the budget next Tuesday and send it on to the Falls Church City Council for final action.

Pace's proposed cuts under the new budget include:

* Dropping five teaching positions. Pace said that if the jobs aren't eliminated by attrition, he would be forced to fire some teachers.

* Combining some grade levels into single classes in the elementary schools, to compensate for fewer teachers.

* Eliminating the swimming program at the high school. Parents and the 28 members of the swimming team have voiced strong opposition to this proposal.

* Abolishing the jobs of some bus aides, who help the drivers supervise children on school buses.

Savings from those cuts would release money for other areas of the budget where costs are rising, Pace said.

Falls Church schools will be forced to pump $15,000 into the lunch program. The federal government has dropped almost two-thirds of its contribution to the school lunch fund this year, forcing the school system to supplement the program for the first time in recent years.

Pace said the schools will raise the lunch prices charged students, although the amount of the increase has not been set. The cost of school lunches jumped 15 cents this year, to 75 cents for elementary students and 90 cents for high school students.

School employes will receive a 5 percent salary increase next school year. The $200,000 cost of the raises will be offset by eliminating the five teaching jobs, Pace said.

Other cost increases include the school system's electricity bill, up $16,800; contributions to employe retirement funds, up $100,000; replacing a school bus and driver education car, $25,000.

The Falls Church school system has about 1,000 students enrolled in two elementary schools and one junior-senior high school.