The Reagan administration, anticipating what White House chief of staff James A. Baker III calls "a long and protracted contest' over the president's scaled-down tax cut proposal, intends to pull out all the stops to get it passed.

Although Baker and White House counselor Edwin Meese III say the president would have preferred a compromise bill to a fight, the contest coming up is the kind that Reagan always has relished the most and in which he has been the most successful.

When he was governor of California, Reagan liked to say the "if he couldn't make the legislature see the light, he'd make them feel the heat" of public disapproval.

Although White House officials have announced no details, it is known Reagan intends to go on national television if necessary to convince a recalcitrant House of Representatives that his tax cut plan is preferable to a Democratic alternative.

In addition, as part of what White House officials call a "vast and far-reaching educational campaign," business groups will be heavily lobbied, wavering congressmen on both sides of the aisle called into the White House for personal chats and conservative Democrats wooed with the promise of at least tacit backing from Reagan in the 1982 elections.

The selling campaign will start today when Baker appears on the CBS program "Face the Nation" and Meese goes on NBC's "Meet the Press." The administration message on both programs will be the same as when the two White House senior officials spoke to reporters in the Roosevelt Room Friday -- namely, that the Democratic alternative is actually a tax increase over a three-year period.

Reagan's latest plan is for a $37.4 billion tax reduction over three years, including a 25 percent personal income tax cut. The proposal calls for a 5 percent reduction in October 1981 and 10 percent cuts in each of the next two years.

The bill, which is nearly $18 billion less in tax cuts than the original administration proposal, also would reduce the marriage penalty, expand savings incentives, cut estate and gift taxes and give incentives for research and development. And it would reduce some of the tax breaks for businesses contained in the original bill.

Both Meese and Baker said an administration victory on the plan in the Democratic-controlled House would be another step toward Reagan's aim of a bipartisan coalition favoring a reduced federal government.

But the aides acknowledge that the president faces some obstacles on the tax plan that he didn't on the budget bill, which he won by a 63-vote margin.

Baker said lines have been drawn harder on the tax bill and some conservative Southern Democrats who supported Reagan won't be backing him this time. He also estimated that from 10 to 15 Republicans are potential defectors. These House members probably will be personally lobbied by Reagan.

The White House has not taken a formal count, but the informal estimate is that between 190 and 195 House members are committed to Reagan's plan, leaving the administration significantly short of the 218 votes needed for passage.

Meese said the administration also was short votes needed for passage of the budget bill at a similar stage in the process.

However, the business community was almost solidly behind the budget portion of the president's economic program. In contrast, there is now unhappiness in segments of the business community over some of the changes the president made in coming up with the plan he will try to sell to the country.

Meese discussed these changes Friday and said they were made for a combination of policy and political reasons. The policy reason for scaling down business tax relief was that the depreciation schedule offered businesses was so generous that in some cases it gave companies greater tax allowances than their actual business expenses.

The White House counselor said also that business benefited from other changes in the new Reagan tax bill, including a reduction of the maximum tax rate from 70 to 50 percent on investment income and the trimming of taxes on oil royalties, which could amount to savings of as much as $500 million for the oil companies.

While no one at the White House would say it, the understanding was that this latter benefit could induce Democratic senators from oil-producing states, especially Sen. Russell B. Long of Louisiana and Sen. Lloyd Bentsen of Texas, to support the president's tax bill.

The White House is confident the Senate will approve a tax bill similar to the version now backed by the president. If so, this would in effect give the administration a double chance to win approval of its program.

Baker said during a luncheon interview at The Washington Post Thursday that he though the administration's insistence on a three-year bill that would cut taxes across the board had improved the Democratic alternative.

If there is a Reagan-type tax bill passed by the Senate, it would then be possible for a compromise to be worked out in conference which would have enough of the features of the administration measure to be signed by the president.

Throughout his political career, Reagan has often signaled a refusal to compromise until the eleventh hour, then signed whatever the legislative body has given him.

This time, however, the president clearly believes he can win far more than half a loaf by taking his case to the country as well as to the Congress.