The Senate Finance Committee served notice yesterday that it may slash taxes this year $10 billion more than President Carter requested.

The committee, preparing a report to the Senate Budget Committee on anticipated tax actions for fiscal 1979, which begins Oct. 1, endorsed a proposal by Chairman Russell B. Long (D-La.) to leave room for up to $35 billion in tax cuts, instead of Carter's proposed net cut of $25 billion.

The House Ways and Means Committee, in a similar report for the House Budget Committee, recommended $3 billion more in tax cuts than the Carter goal.

Final agreement in Ways and Means was held up for hours while the committee wrangled over whether parts of the cut should be in Social Security rather than income taxes.

The House committee, on a motion by Rep. Joseph L. Fisher (D-Va.), finally decided to put the Social Security issue aside for later consideration --(D-Ore.) satisfied Rep. Richard A. Gephardt (D-Mo.) and several others that he would allow a Social Security tax cut to be proposed as part of the pending general tax cut bill under certain conditions.

Ullman explained that he opposes cutting Social Security taxes and simply pumping income tax funds into the Social Security system to make up for the loss. He would rule such an option out of order as part of the tax bill, he said. But he would allow consideration of a plan to take some part of the Social Security system -- for example, Medicare -- off the payroll tax and fund it permanently from general revenues.

House Democrats anxious for a Social Security cut said they had collected enough signatures last night to force a Democratic caucus and press Ways and Means to act on a Social Security reduction. New Social Security taxes were voted only last year, but constituent complaints are so loud that many members now want to block the increases and use general revenues to help support the Social Security system.

Ways and Means rejected such a demand, 19 to 18, on Wednesday, but Rep. Abner Mikva (D-III.), William A. Steiger (R-Wis.) and others want to bring it up later and now seem to have a pledge from Ullman that he will allow some form of reduction to be considered.

The Senate Finance Committee action in raising the possible tax cut to $35 billion may set off a fight when the budget Committee turns to drawing overall tax and spending targets this spring. Many members of the Budget Committee were said to feel that under present economic conditions, the President's $25 billion tax cut would be for a private school/colsubstantially increased in view of the $61 billion proposed budget deficit unless the economy shows signs of sagging.

The Finance Committee did not specify where the extra tax cuts might be made nor to whom they would go. Technically, it could ignore Carter's specific proposals and give the cuts to entirely different groups (he has proposed that three-quarters be individual income-tax cuts primarily for lower-income persons, and that the rest go to corporations). And it could vote less than $35 billion when it comes to actually writing tax legislation.

However, committee discussion made clear that $1.2 billion of the extra cut would be for a private school/college tuition tax credit already approved by the committee. Carter opposes this. Another $5.1 billion would be for energy tax cuts in the Senate version of the stalled energy bill. If the energy cuts should not materialize, the authority could be used for cuts elsewhere.

Republicans John C. Danforth (R-Mo.) and William V. Roth Jr. (R-Del.) argued that the president's proposed $25 billion tax cut isn't enough to provide equity to middle-income taxpayers, and that inflation is lifting many low income taxpayers into higher tax brackets.

Roth said he may offer an amendment that could give individual taxpayers the entire $10 billion extra for which the Finance Committee's $35 billion ceiling leaves room. This would be accomplished by a flat 10 percent reduction in income tax rates for all taxpayers, instead of the selective individual cuts asked by Carter.

In addition to up to $35 billion in new tax cuts, the Finance Committee also recommended that $9 billion in existing tax cuts, expiring this year, be extended another year. The Carter and Ways and Means proposals also allow for this.

The finance Committee also notified the Budget Committee that it expects to add several spending items to the budget, including $400 million in relief for the states' welfare costs and $200 million more for the social services grants program, bringing it to $2.9 billion. Ways and Means has approved these items to. The committee also indicated it doen't expect hospital cost containment bills to pass in time to have an impact this year.

It also voted, 7 to 6, with absent members to be polled, to continue the countercyclical revenue-sharing program at $1.5 billion for next year, instead of Carter's requested $1 billion.