Fairfax City will hold a bond referendum April 12 to ask voters to approve $6.2 million for repairs and modernization at six of the city's seven schools, including the controversial addition of gymnasiums at four elementary schools.

The school bonds would add an estimated 9.2 cents to the city's current $1.15 real-estate tax rate, said City Manager Edward A. Wyatt, or just about the nine cents by which the City Council last year cut the tax rate.

The council is struggling with a $2.2 million budget shortfall, and at last week's meeting members estimated the property tax rate will have to increase four to 10 cents even without the school bonds.

The shortfall has come primarily because of last year's apparently over-generous nine-cent tax cut, which used up a temporary $1.67 million city surplus, and by reduced sales and property tax revenues in the past year's stagnant economy, Wyatt said.

Most of the school bond money would be spent on renovation of the 20-to-30-year-old school buildings, including a new roof for Fairfax High School and new wiring, heating and other mechanical improvements at Lanier Intermediate School and four elementary schools.

The city's fifth elementary school, John C. Wood, will close at the end of this year and will be sold or used for other city activities.

The last bond issue, in 1979, was for $2.6 million to buy four of the schools from Fairfax County.

"What we're proposing now would not do everything the county does in its regular school renovation program , and even the gyms are very modest, without wooden floors or bleachers," school board member Dale Gaddy said. The gym additions would cost $1.2 million, or about $300,000 each.

Twenty years ago, few elementary schools had gyms or physical education programs, said Gaddy, an official with the National School Board Association. "But in the last couple of decades there has been increased emphasis given to physical development of schoolchildren, . . . and without gyms there often can be no volleyball, basketball and other sports for students because of winter and bad weather."

Elementary students in the city now use converted classrooms for exercises but have insufficient room for ball games and can exercise only in small groups, Gaddy said.