President Carter intends to ask Congress to enact an increase in the gasoline tax despite the lawmakers' overwhelming rejection of his 10-cent-a-gallon oil import fee earlier this month, his chief adviser said yesterday.

Speaking on the CBS television show "Face the Nation," the President's adviser for domestic affairs, Stuart E. Eizenstat, also affirmed that the administration would move toward tax cuts next year to stimulate productivity, capital investment and savings. He admitted balancing the federal budget in fiscal 1981 would be "difficult though not impossible," but declined to speculate how large the deficit is likely to be.

President Carter orginally proposed the gasoline tax back on March 14 at the same time he announced an oil import fee of $4.62 a barrel. The fee, also amounting to a tax of 10 cents a gallon on gasoline but not other petroleum products, was designed to reduce imports by 100,000 barrels a day in one year and by a quarter of a million barrels a day within three years.

The oil import fee, which would have raised $10.3 billion annually, ran into opposition from members of Congress who contended it was primarily a revenue raising ploy to balance the budget. Moreover they said the effect was inflationary though not serious enough to promote conservation. Last month a coalition of gasoline marketers, citizens groups and five congressman obtained a federal court order to prevent the fee from going into effect May 15. The adiministration appealed. p

Then in early June both houses of Congress voted overwhelmingly to kill the important fee despite threats of a presidential veto. Carter did veto the measure and the legislators did override by large margins. It marked the first time in nearly 30 years that Democratic president had been overriden by either a Republican or a Democratic controlled Congress.

The President served notice that he intended to propose a 10-cent-a-gallon increase in the current 4-cent-a-gallon Federal gasoline tax eventually to replace the import fee. However, it was assumed that when Congress rolled back the import fee, Carter would drop the gasoline tax plan as well.

Yesterday, just 16 days after the measure was overridden, Eizenstat said the president would follow through on his March 14 announcement to send to Congress a request for "a specific gasoline tax, in the same amount exactly, which will replace the conservation fee."

In that statement Carter pledged not to "use these revenues to balance the budget or as a substitute for necessary spending cuts. . . These revenues will give the budget, which will be balanced, a margin of safety, ensuring that it will remain in balance if conditions or estimates change in a way that we do not anticipate," he declared. Since that speech both conditions and estimates appear to have changed in a way the administration did not anticipate as the country has fallen very rapidly into a recessionary spiral.

Eizenstat said yesterday the gas tax was intended to have the same effect as the gas fee.