The "Selling of the Tax Cut," a Republican extravaganza orchestrated at the White House and paid for out of the party's bulging coffers, begins in earnest next week.

Republican leaders announced on Capitol Hill yesterday that they will spend up to $500,000 on a nationwide radio advertising campaign calling for passage of President Reagan's tax-cut proposal and rejection of an alternative measure sponsored by the House Democratic leadership.

Meanwhile, the proposal they will be trying to sell was undergoing modification at the White House yesterday. House Republican leaders said Reagan had agreed to substantial changes in his legislation to lure undecided votes from the Democrats' bill.

The GOP radio blitz, which begins Monday and is to be paid for by the Republican National Committee and the Republican Congressional Campaign Committee, will include commercials on nationwide radio networks and other messages to be aired in targeted congressional districts represented by wavering Democrats, the GOP leaders said.

While his program is being pushed on the airwaves, the president also is expected to step up his activities next week, when the tax debate is scheduled to move to the House floor.

To underscore his determination, Reagan told a group of state legislators and local government officials at the White House yesterday that there should be "no doubt that I'll go anyplace, any time, to ensure that the working people of this country get their first real tax reduction in nearly 20 years."

The president is scheduled to speak Wednesday in Atlanta to a conference of state legislators, where he is expected to push his tax plan.

Augmenting these efforts to enact the Reagan tax plan will be at least two others. The National Conservative Political Action Committee (NCPAC) has announced that it has targeted 13 Democratic members of Congress for advertising attacks beginning next week unless they reverse their opposition to the president's tax proposal.

John T. Dolan, the head of the conservative group, threatened to spend as much as $500,000 in the negative advertising campaign.

Dolan said the Democratic members of Congress, selected because they are in what he called precarious districts and have not endorsed Reagan's 25 percent cut in personal income taxes over three years, are George E. Brown Jr. of California, Ray Kogovsek of Colorado, William R. Ratchford of Connecticut, Paul Simon of Illinois, Neal Smith of Iowa, Jamie L. Whitten of Mississippi, James J. Florio of New Jersey, Thomas J. Downey of New York, Stanley N. Lundine of New York, Stephen L. Neal of North Carolina, Bob Edgar of Pennsylvania, Thomas S. Foley of Washington and Les Aspin of Wisconsin.

The other effort for the tax plan is to be mounted by Alabama Gov. Fob James, a Democrat who supported Reagan last fall. With the assistance of the White House advance office, James is to take off next week on a three-day tour of seven southern states urging passage of the Reagan program.

A James aide said the state of Alabama will pay for the trip.

The most ambitious of the sales jobs is the radio campaign unveiled yesterday by Rep. Dick Cheney (R-Wyo.), chairman of the House Republican Policy Committee, Rep. Guy Vander Jagt (R-Mich.), chairman of the GOP Congressional Campaign Committee, and Richard Richards, chairman of the Republican National Committee.

In addition to the radio campaign, Rep. Standord E. (Stan) Parris (R-Va.) announced at the same news conference that his "Committee for Economic Recovery" will spend about $30,000 on radio and a television commercial to be braodcast in the Washington area.

The television commercial depicts House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) as a deceitful "Christman speaker" who offers "this nice big package that looks just like a tax cut" but isn't.