The Senate Finance Committee served notice yesterday that it may seek to cut taxes $6 billion more than President Carter has proposed in the fiscal year beginning Oct. 1.

One possible extra cut the committee briefly discussed is a $250 college tuition tax credit that has been proposed by Sen. William V. Roth (R-Del.) and others. It would cost about $1 billion a year.

The Senate approved such a credit as part of last year's tax bill, but the provision died in conference with the House.

The Finance Committee is drafting a letter to the Senate Budget Committee as an early step in the new congressional budget process. The Budget Committee will take this and other letters into account in drafting its spring budget resolution, setting federal tax and spending targets for the fiscal year ahead.

The Financil panel made clear yesterday that it wants these targets get in such a way as to give it maximum latitude in future legislation.

Thus on the one hand it reversed a decision it had made Tuesday, to tell the Budget Committee it might increase Social Security taxes about $1 billion next year.

One reason for the reversal was a fear that if the Budget Committee included the $1 billion in its revenue projection, then the Finance Committee would be bound to raise the money. By not making a commitment now, the committee retains more freedom later.

A second reason for the change of heart was a telephone call yesterday morning from Health, Education and Welfare Secretary Joseph A. Californo Jr. to Finance Committee Chairman Russell B. Long (D-La.).

Long said Califano told him the administration opposes increasing the automatic increase that occurs each year in the maximum amount of wages subject to the tax. Among other things, the administration thinks a Social Security tax increase would depress the economy just when it needs stimulating.

Social Security benefit payments are now outrunning Social Security tax collections each year, and the system is eating into its reserve funds. But the administration says it thinks there is enough in the reserve funds so that no tax increase decision needs to be made this year.

The Finance Committee was also acting to maximize its future freedom when it said it might want to cut taxes $6 billion more than Carter next year. The more tax-cutting room the spring budget resolution leaves it, the happier the Finance Committee will be. Then if it votes a big tax cut, no one will be able to accuse it of breaking the budget and increasing the federal deficit.

The $6 billion figure was pretty much plucked from the air. The President's tax cut proposals will not reach the Senate until next week, and the Finance Committee is not sure what it may decide to do with them - nor what decisions it may make on any subsequent tax bills this year.

The Roth proposal is one possibility. Long suggested a liberalization of the present child-care credit as another. And Republicans favor various across-the-board tax reductions; to stimulate the economy they would rather cut taxes more, and increase spending less, than the President has proposed.

In general, the Roth plan would let parents reduce their taxes by $250 for each child enrolled in college, as an offset to tuition. Self-supporting students would be allowed similiar credits.

In the House, meanwhile, the Rules Committee yesterday unanimously cleared for floor action today the version of the President's tax cut proposals approved two weeks ago by the Ways and Means Committee.