Alexandria City Manager Douglas Harman yesterday proposed a 7 per cent increase in the city and schools operating budget for 1977-78 and recommended that the city increase its real estate tax rate by eight cents, to $3.58 for each $100 of assessed value.

Harman's $79.8 million budget, presented to the City Council yesterday morning, includes "enhanced and expanded" services in some departments, but also eliminates nine city positions, some of which may involve firings, according to the city budget director.

Harman said that this eight-cent tax increase proposal was necessary to balance the budget and continue the same level most city services.

Under Harman's proposal, the real estate tax for a $60,000 home in Alexandria would increase $24 the coming fiscal year, from $1.050 to $1,074.

Harman said the tax increase would raise $850,720 for the city.

Alexandria Mayor Frank E. Mann said yesterday he doubts the city Council, which makes the final decision on the budget, will go along with a tax increase.

Noting that several council members said they would not support tax increase during their election campaigns, he said he is certain he and the council will "get some sharp pencils" and find the necessary money without raising taxes.

The city manager said several program expansions approved earlier by the council are in the budget proposal.

They include a dog control program, creation of a third rescue squad, nine new positions on the city jail staff, a noise pollution control program and the alcohol prevention program in the city's schools. He also asked for additional staff members in some recreation centers.

Harman also recommended that the city stop paying for three community youth worker positions, the extended day care program under which the children of working parents are supervised after position in the Alcohol and Drug Abuse control program. He said all three services can be provided by other agencies.

On salary and benefit adjustments, the city manager recommended a general increase of 4 per cent. City Councilman Donald C. Casey suggested yesterday one way to balance the budget without a tax increase would be to keep salaries at the present level.

The Alexandria proposal calls for the smallest operating budget increase in the Northern Virginia suburbs. Earlier this week, Fairfax County Executive Leonard Whorton proposed a $315.9 million budget for day-to-day operations of the county, up 11.7 per cent.

Arlington County Manager W. Vernon Ford last week recommended a $107 million budget that is 9 per cent above this year. In Prince William County, County Executive Clinton B. Mullen has suggested a $26.7 million operating budget, an increase of 11 per cent over the current budget and which does not yet include the school budget.

Only in Alexandria and Prince William County were real estate Tax increases suggested.

Harman also proposed a $21.8 million capital improvement budget which includes city funds of $15.3 million with the rest coming from federal, state and other sources.

Harman said that among the capital construction projects expected to be completed in 1977-78 are the Seminary road Health Clinic and the George Mason and William Ramsay Elementary School multipurpose rooms. Funds will also be used to start construction of the new city courthouse, he said.

Harman said by adding city and school operating expenditures, city capital improvement funds, state, federal and other contributions, the total budget is $106.6 million.

The Alexandria school system accounted 41 per cent of Harman's proposed budget an increase of 5.78 per cent over this year and the single largest expenditure. The Metro operating deficit was 4.2 per cent of the operating budget.

Transportation and environmental services consumed 7.3 per cent of the budget and 14.3 per cent went for the protection of persons and property, such as police and fire aid. The remaining money is spread among various services such as recreational activities and staffing of agencies.

On the revenue side, general property taxes contribute 55.5 per cent of the city funds. Other local taxes were second, at 15.1 per cent. The rest of the money comes from various other sources, such as licenses, fines and school revenues.

After holding numerous work sessions and a public hearing, the City Council is expected to make a final decision on the budget by May 9, Harman said.