The new chairman of the Republican National Committee prodded President Reagan yesterday for an early "signal" of candidacy and announced that he is going to convene a "broad cross-section" of women to tell the GOP how to regain female support and close the gender gap.

Frank J. Fahrenkopf Jr., of Nevada, who was installed last Friday at the party helm, told reporters that a formal announcement from the president could wait until July 4, but that the party organization and other potential candidates need a signal of Reagan's intentions "quite soon."

Asked how soon that might be, Fahrenkopf replied, "Tomorrow would be nice."

His statement seemed certain to feed growing demands from Republican activists for an early decision on a second-term candidacy. Such expressions were heard from many officials at last week's meeting of the Republican National Committee, which formally endorsed Reagan and Vice President Bush for reelection.

White House communications director David R. Gergen commented yesterday that Reagan has already given "a number of signals" of his interest in running and "has no current plan" to go beyond them.

Other White House aides have spoken of a possible Labor Day announcement of his plans.

Meanwhile, Fahrenkopf said that as a "top priority" for the party, he had asked national committee co-chairman Betty Heitman to put together a series of "forums . . . with women of all kinds, to listen to their problems and see how we can respond."

For the last couple of years, polls have consistently shown lower support for Reagan and the Republican Party among women than among men. Fahrenkopf said yesterday he believed "the gender gap is real. Single women and single heads of household in particular have a perception . . . that this administration and party do not care about their problems."

He said he had asked Heitman on Monday, his first day at headquarters, to plan a series of meetings where women of all kinds, "not just the Republican women's club types," can meet with women in the Reagan administration and Republican women elected to state and local office and talk about "the steps we need to end discrimination." Staff aides said they hoped to have the first session in Washington within two months.

Noting the "length and complexity" of the presidential campaign, Fahrenkopf said that if Reagan is going to run, "work should get started as soon as possible. And if we're not going to have a Reagan reelection campaign, other candidates" need and deserve the opportunity to test their support.

Fahrenkopf said that Reagan's designation of Sen. Paul Laxalt (R-Nev.) as general chairman of the Republican Party was "a good signal to me" that Reagan will run. But he said Reagan's approval of fund-raising and organizing efforts on his behalf or a word of caution to "members of Congress who might be looking to make the race" would be more convincing.

He declined to tab Bush as the favorite in such a contest, saying he was sure the vice president would be challenged by several senators, including Laxalt, if a vacancy occurred.

One of the people he mentioned, Senate Majority Leader Howard H. Baker Jr. (R-Tenn.), told reporters at a separate meeting that he expected Reagan to run again but did not feel "boxed out" of running himself in 1984 if Reagan steps down.

Baker said one reason he is retiring from the Senate next year is that "the Senate is an overrated place from which to run for president." He said the "legislative mindset" is "more of a liability than an asset" in campaigning around the country and the time demands of Senate service are almost incompatible with a presidential race.