With angry Republicans booing from the sidelines, House Democrats confronted President Reagan yesterday with a divide-and-conquer strategy intended to defeat his latest budget offensive before it gets rolling.

The showdown is expected to come today, and Reagan lost no time in mounting a counteroffensive against the Democratic challenge.

In a statement issued from the White House as the president was homeward bound to California, Reagan accused the Democrats of engaging in "backroom politics" and called on the American people to demand that his program get a fair vote in the House.

"I want the American people to understand that if they want to bring real change in Washington, if they want real reductions in spending, this is the time to speak up," said Reagan. "This is the time to be heard." [Details on Page A3]

In announcing the Democrats' strategy, Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) said the Democrats will allow the House to vote on Reagan's proposals for even deeper spending cuts than House committees approved -- but not in the form that Reagan and the House Republicans want.

Rather than give Reagan the single up-or-down vote he sought for his package of proposals, O'Neill told reporters that House members will have to vote on six pieces, stripped of all provisions to sweeten the cuts by restoring some money for popular programs. Thus, in order to support Reagan, House members will have to cast six record votes against some popular programs rather than a single vote in favor of the president's program.

Majority Leader James C. Wright Jr. (D-Texs. also told the House that the Democrats will push to finish work on the spending cuts before Congress quits late in the week for its July 4 recess, rather than postponing final action on the cuts until after Congress returns.

In the floor action today, House Republicans will try to force a full up-or-down vote on Reagan's proposals. But they were so stung and thrown off balance by the Democratic ploy that they said they may wash their hands of the amendments -- which they said now amounted to "Democratic amendments" -- if they lose the fight to vote on the Republican proposals as a package.

Today's showdown will come on a procedural point, involving approval or rejection of the Democratic strategy to vote on the Republican program in pieces. But it is expected to be the most dramatic thus far in the 97th Congress. Even yesterday, emotions ran high. Republicans booed and hissed as Wright spoke, with Democrats responding in kind when Republicans arose to denounce them.

The Democrats' strategy gives the president's foes the advantage in several important ways. Members would be voting on specific spending cuts, such as college loans and increases in federal workers' pensions, rather than for or against Reagan's overall economic program, which is vastly mofe popular than some of its specific provisions. Moreover, Reagan may not have time to duplicate the effective lobbying campaign he waged to win the first round of the budget battle last month.

"They don't want the president to speak out. They're afraid of him," complained Rep. Delbert L. Latta (R-Ohio), ranking minority member on the House Budget Committee. But Latta also conceded that the Democrats' strategy may work. "Unless the House has changed more than I think it has, you'll find most of the [president's] amendments will go down," he told a reporter.

Reagan, however, pursued an intensive lobbying campaign from long distance, sending telegrams to all 190 House Republicans and the 63 Democrats who supported him on the budget last month. He "urgently" appealed for a vote against what he called the Democrats' "gag rule." Aides who remained at the White House worked into the night on the effort.

The lobbying campaign was aimed at conservative Democrats who helped Reagan win his earlier budget fight, setting targets for spending cuts. But senior Reagan aides conceded they were at a disadvantage fighting over rules, where party discipline is usually strongest, instead of having their showdown on the substance of Reagan's program. "It's really tough," said one aide. "I don't know how much can be done." But House Minority Leader Robert H. Michel (R-Ill.) characterized the fight as "winnable."

If the Democratic strategy works, it will be one of the rare occasions since Reagan took office that the Democrats were able to outmaneuver him. But it will be more of a political, psychological and symbolic victory than anything else because Reagan has already won most of the budget cuts he wants from both houses.

Moreover, as the Democrats were flexing their muscles in the Democratic-controlled House, the Republican-run Senate continued to move inexorably toward approval of its version of the huge budget "reconciliation" bill brushing aside Democratic amendments to restore money for programs ranging from energy conservation to economic development.

At issue in both chambers are spending cuts recommended by committees to meet budget targets approved by the two houses last month, including heavy cuts in domestic programs amounting to $39.6 billion in the Senate and $37.7 billion in the House.

Reagan has no quarrel with the Senate's work but contends that the House committees fell short of their quotas of real savings by approving sham cuts in some areas and failing to crack down sufficiently on open-ended entitlement programs.

So the Republicans, with Reagan's blessing, proposed their own version of cuts for money programs, from college loans to Medicaid to subsidized housing, contending their package would save $20 billion more than the commmitte's proposals over three years. They also proposed including a modified version of Reagan's plan for consolidating many federal programs into block grants to the states, with fewer strings as well as less money from Washington.

As finally approved by the Rules Committee, the Republican alternative will be carved up so that the House will vote separately on:

Tightening eligibility for receiving food stamps, including barring anyone with an income in excess of 130 percent of the government-established poverty level from receiving stamps. The committee-approved proposal would simply put a "cap" on food stamp expenditures, leaving eligibilty changes to be enacted in separate legislation that is currently on its way to the House floor.

Making deeper cuts in child nutribution programs and imposing a means test for government-guaranteed loans to college students by limiting loans to students' actual need. There is no means test for college loans now, and the committee proposal does not include one.

Reducing the number of new and rehabilitated subsidized housing units from 176,000 to 162,000 and increasing tenants' share of rental costs from 25 percent to 30 percent over the next five years.

Dropping one of the two cost-of-living increases that federal retirees receive every year in their pensions. The Democrats planned initially to include the Republicans' proposal for reducing next year's pay increase for federal workers from 5.8 percent to 4.8 percent, but dropped the idea because they figured it might lose them votes.

Eliminating the $122 monthly minimum benefit under Social Security as of April, 1982, and providing for faster phase-out of Social Security student benefits. Aid to Families with Dependent Children would also be reduced in several ways, including making strikers ineligible for welfare benefits. States would be permitted to reduce welfare payments by the value of food stamps and housing subsidies that a recipient receives.

Imposing a "cap" on Medicaid payments to states, coupled with greater flexibility for states to reduce Medicaid costs.

The plan approved by the Rules Committee also would give the House a chance to vote on whether the committee proposals should be redrafted to include Reagan's block-grant idea.

In his attack on the Democratic leaders, Reagan charged that they were pursuing a strategy that would "once again allow special-interest groups to triumph over the general economic interests of the nation" and described the committees' proposals as "a scheme . . . that would effectively sabotage our attempts to cut federal spending."

Characterizing the budget contest as the "best opportunity in years to achieve real change in this country, Reagan said, "We just can't surrender it to backroom politics in the halls of Congress."