The chairman of the Senate Finance Committee said yesterday there is growing support in the Senate for a three-month postponement of next year's scheduled cut in income taxes, as one alternative to the additional budget cuts that have been proposed by President Reagan.

Describing the pressure for the deferral as "substantial," Sen. Robert J. Dole (R-Kan.) said he has told White House officials: "You guys better get with it or you are going to have a problem on your hands."

At a breakfast with reporters, Dole said he personally opposes tinkering with the recently-enacted tax cuts in an effort to save money, but he added that he did not "open the door," and consequently it is not his role to "slam it," meaning he will not take the lead in opposition.

Administration officials have strongly rejected proposals for postponing the tax cut--a 10 percent rate cut in rates in all brackets scheduled to take effect next July 1. Earlier this week, Treasury Secretary Donald T. Regan, for example, declared in a speech in San Francisco at "we have no intention of deleting, delaying, or backing off from the three-year, 25 percent tax rate cut," in which next year's scheduled 10 percent would be the second installment.

The administration has, however, called on Congress to produce at least $13 billion in new spending cuts in fiscal year 1982 and to pass $3 billion in tax increases for the same year. Senate Republicans are openly balking at the budget cuts and are considering an assortment of other methods to hold down the deficit.

One of the alternatives they are considering is increases in federal excise taxes, particularly the "sin" taxes on tobacco, beer, wine and distilled alcohol.

The congressional staffers have worked up estimates showing that a doubling of the cigarette tax to 16 cents a pack would raise $4.8 billion over three years, of which $1.2 would be raised in 1982; a doubling of liquor taxes to $21 a gallon would produce $1.2 billion in 1982 and $5.5 billion over three years; and the raising of beer and wine taxes to $18 a barrel and 34 cents a gallon respectively would produce $0.9 billion in 1982 and a total of $3.5 billion through 1984.

Similarly, postponement of next year's scheduled 10 percent rate cut from July 1 to Oct. 1 would produce $7.4 billion in fiscal year 1982 and about $1 billion in 1983. Postponement of the 1983 rate cut for three months would raise the revenue savings in 1983 to $8.3 billion to add $0.7 billion in 1984.

As alternatives to making new and deeper cuts in such programs as food stamps, health services, Amtrak and school lunches, the excise tax increases and postponement of the individual rate cuts are politically attractive.

Dole, noting that Congress faces "hard choices" in the drive to reduce the deficit, pointed out that "we are known for taking the easy way out." In distributional terms, the excise tax increases, which are similar to sales taxes, would fall most heavily on persons with low to moderate incomes.

The Reagan tax "enhancement" proposals are all sure to provoke strong opposition from influential interest groups including the insurance industry, organized labor and defense contractors.

While not advocating the delay of the individual rate cuts, Dole pointed out that it might be possible to package this with a parallel delay in cost-of-living increases for such programs as federal pensions, school lunches and food stamps. "I don't recommend it," he said, but he said in political terms this tactic of reducing the deficit would be less "painful."

Referring to one of the tax increases proposed by Reagan--the administration prefers to describe them as tax "enhancement" proposals--Dole said there is little or no support for one of them: ending a network of tax credits for persons and companies that shift to using solar and other renewal energy sources.

Dole made it clear that congressional action on significant tax changes will not take place until next year. In the case of the excise taxes, any delay past Jan. 1, 1982, would function to reduce revenues from whatever increases are approved.