Republican presidential hopefuls begin to run the conservative gauntlet in earnest this week, reflecting the calculation of political strategists that the GOP's right wing is up for grabs for the first time since Sen. Barry Goldwater (R-Ariz.) won the nomination in 1964.

The intensified dueling for support from the right begins Thursday when Vice President Bush and Rep. Jack Kemp (R-N.Y.) are scheduled to appear at a dinner honoring the 23rd anniversary of New York's Conservative Party.

Kemp initially turned down the invitation but changed his mind after learning that Bush had agreed to speak at the dinner. Bush will be the "guest of honor" while Kemp's status has been defined as "special guest."

The New York dinner will be followed by the 13th Annual Conservative Political Action Conference (CPAC) here starting Jan. 30, a high visibility event that all prominent GOP contenders, except former senator Howard Baker Jr. (R-Tenn.), are expected to attend.

Between those two events, Bush will attempt to firm up support on the religious right as keynote speaker Jan. 24 at the Rev. Jerry Falwell's Liberty Federation, as the Moral Majority has been renamed. The day before the dinner, Kemp is scheduled to speak to antiabortion protesters on the Capitol steps.

All the GOP potential candidates, from the Rev. Marion G. (Pat) Robertson to Senate Majority Leader Robert J. Dole (R-Kan.), have participated in lengthy interviews published this month in Conservative Digest. The magazine assigned such conservative leaders and journalists as Connaught Marshner, Howard Phillips and Patrick B. McGuigan to press the prospective candidates on school prayer, abortion, the gold standard, affirmative action and other issues.

In the months ahead the candidates will be invited to "cattle shows" before the antiabortion National Right to Life Committee in Denver and the National Conservative Political Action Committee in Dallas. And candidates will make numerous individual appearances before antiunion, pro-gun and other conservative groups.

David Keene, who is running CPAC, said the early and intense courting of the political right is taking place because "no one has grabbed the heart of the movement. . . . They have to do it because no one owns the conservatives this time."

In effect, just as the withdrawal of Sen. Edward M. Kennedy (D-Mass.) has opened up competition for the left within Democratic presidential politics, the coming retirement of President Reagan has opened competition for the right.

The result is avid competition for conservative support among the candidates. Bush is the frontrunner for the nomination. And all the rest are seeking to erode his status.

"Kemp is of the conservative movement and always has been," Kemp press aide John Buckley said. "The vice president is a national leader of the Republican Party and an extremely good and loyal vice president. His roots, however, are not as a movement conservative."

Craig Shirley, a consultant to the Bush campaign, countered: "Kemp has been taken aback by the fact that much of the conservative movement has not been knocking down his door. Kemp thought he was going to inherit the Reagan mantle, but it's wide open."

The early and sharp competition for support from the right has produced an atmosphere in which motivations are under constant scrutiny.

Keene, who worked for Reagan in 1976, Bush in 1980 and who now backs Dole's bid, said, in partial jest, that among members of the American Conservative Union, which is sponsoring the CPAC event: "One-third think I'm a secret agent for Bush, one-third think I'm a secret agent for Dole, and the rest think I have no moral principles at all."

One issue that has become a focal point of the jockeying for conservative backing is abortion. "I think it's rather significant that the major names out there Bush and Kemp are accepting invitations to come to our annual meeting," Jack Willke, president of the National Right to Life Committee, said. He pointed out that it is "the first time the vice president has come to speak" although not the first time he has been invited.

Abortion has been one of the more difficult issues for Bush, who, as recently as 1980 was considered a moderate on abortion rather than a hard-line opponent. "There has been some evolution in my views on abortion, given the enormity of the number of abortions. I find it alarming and deeply offensive," he told Conservative Digest.

Kemp, who has had a consistently antiabortion record, was criticized, however, last summer by such new right leaders as Paul Weyrich, head of the Free Congress Foundation, and Howard Phillips, president of the Conservative Caucus, for playing down the issue and other social questions.

Kemp began to speak out more often on abortion, prompting Weyrich to say: "This just goes to show that the market system works."

While all the candidates are jousting for the chance to show off their credentials to a wide range of conservative groups, some offers are turned down.

Buckley, Kemp's aide, said Kemp received an invitation Jan. 17 to speak Jan. 24 at Falwell's Liberty Federation meeting. Kemp turned down the invitation for two reasons: "because he will be on his way to the Super Bowl" and because his schedule is determined six weeks in advance.

In addition, however, Falwell has already endorsed Bush for president in 1988.