District financial institutions are likely to soon get the tax relief they have sought for years, even though it means getting higher tax bills this year.

This ironic situation comes about as the result of the D.C. City Council's approval this week of a bill to tax the net rather than gross incomes of banks and savings and loan associations. They also will be subject to personal property taxes.

But, because the city has a serious budget deficit, the council arranged to phase in the relief over a three-year period without loss of revenues at a critical time. In return for eventual relief, banks and thrifts will pay an additional $3.6 million in D.C. taxes in fiscal 1981.

Dale L. Jernberg, president of the D.C. Bankers Association, said members would not object to the legislation "in view of the overall quality of the proposed bill, even though it results in a continuation of the inequitable tax burden imposed on financial institutions for up to three years."

Banks and thrifts had eagerly sought relief from what they claimed was a much higher tax burden placed on them than on nonfinancial businesses in the District and on financial institutions in almost all states.

By taxing gross instead of net income, the District last year collected $2.2 million from four S&Ls that had combined losses of $9.8 million. Three out of 16 D.C. associations paid gross earnings taxes in excess of their incomes. This year it is estimated savings and loans will have to pay $8.1 million in gross earnings taxes while suffering losses of $7.4 million.

The legislation was unanimously approved by those council members present at the preliminary action Tuesday. Final action is scheduled for June 17. Mayor Marion Barry, who originally supported the bill, is expected to sign it. Without opposition in Congress it would become law 30 days later.

Instead of a gross earnings tax of 5 to 6 percent on banks and 2 percent on thrifts, financial institutions would thereafter pay a franchise tax of 9.9 percent on net income plus a personal property tax of $2.83 per $100 of assessed value on items like equipment. However, during the first year the institutions must make up the entire difference between the tax liabilities under the two systems. Subsequently the difference will be reduced to two thirds and finally one-third.

This year, for example, a bank with a gross earnings tax of $150,000 and a franchise tax of $100,000 would have a tax liability of $150,000. If, on the contrary, the tax on gross earnings were $50,000 and the franchise tax were $100,000, the bank would have to pay $100,000.

In the second year the tax in the first example would go down to $133,000, but would remain at $100,000 in the second example. The District would reap an additional $554,000 in fiscal 1982.

After that, the relief would result in revenue losses to the District of $2 million in FY 1983, $4.3 million in FY 1984 and $7.2 million in 1985.