The White House is engaged in a many-sided campaign to rescue an image of leadership which some of President Reagan's most trusted advisers say they fear is slipping from his grasp.

Talk of "bold new steps" that the president can take to regain the economic initiative dominate staff and Cabinet discussions at the White House.

A long menu of proposals ranging from the flat income tax to new initiatives to cope with unemployment are getting urgent attention. The president has been urged to make a nationally televised speech next week to convey what one called "a bold new message of leadership," but White House officials said last night that this is not likely because it would crowd the State of the Union message the president must deliver on Jan. 25.

Reagan's close advisers say they believe that it is urgent for him to show that he has a plan for the nation and that he is in control of his dissonant administration.

Recent signs of White House disarray have prompted discontent among loyal Reaganites in Congress and increasing concern among the president's top advisers.

The president is described as particularly annoyed--one official said he "blew his top"--about reports that he has delegated most of his key decisions to a multi-voiced staff that seems to agree only on the necessity of talking Reagan out of his most cherished views.

This growing annoyance at being portrayed as a detached, out-of-touch president being manuevered into reasonable positions by a staff that sometimes does not seem overly respectful of Reagan's abilities was one reason that Reagan expressed outrage this week about White House "leaks."

His concern about the leaks is longstanding, but he is bothered even more by the notion that his staff is determining the presidential agenda.

White House counselor Edwin Meese III yesterday disputed the accuracy of this perception, saying that "the president was continuing to make the tough decisions." He said that any questions about Reagan's leadership would be answered by the budget and the State of the Union address.

"These are the two major blueprints that will show the continuing impact of the president's leadership," Meese said.

Deputy chief of staff Michael K. Deaver, discussing Reagan's budget decisions and the appointment of two women to the Cabinet, said, "I don't see this as a hype to recapture leadership. I see this as a normal exercise of presidential leadership in solving problems."

Despite the official optimism, administration officials acknowledged that the president is slipping in public esteem. This perception has been reinforced by surveys taken by Republican pollster Richard B. Wirthlin that show a steady decline in the president's support.

Public polls convey a similar message. The president's overall approval rating slipped from 45 to 38 percent in the past five weeks. This figure is lower than those of other surveys, including Wirthlin's, because the Harris Survey counts those who think the president is doing "only fair" as negative responses. But the trend line is the same.

All the polls, public and private, show that Reagan has a continuing problem on what politicians generally refer to as "the fairness issue," the belief that administration policies have been especially harmful to the poor, minorities and the disadvantaged.

Discussing the unemployment issue, one administration official said recently, "We've got to do more than just sit here and talk about it as another statistic in the economic picture. It's not just a statistic; it is something affecting a substantial number of Americans in this country who are now starting to lose their houses and their cars and their kids' lifestyles and a whole variety of other things."

For this reason, some administration officials want the president to show that he does have understanding and concern by taking such steps as visiting an unemployment office or a relief center in a middle-class community.

Some of these officials acknowledge that Reagan may not understand the depth of the jobless problem, but they insist that he is sensitive and would respond to first-hand witnessing of hardships.

Next week, when he goes to Chicago to speak at a mammoth fund-raiser for Sen. Charles H. Percy (R-Ill.), the president also will make a return visit to Providence-St. Mel, a largely black parochial high school on the west side of Chicago. There will be many more such events in the month ahead, if some of the president's advisers have their way.

Only a month ago the president's economic advisers were cautioning Reagan that Congress would reject a budget with massive deficits that made additional huge cuts in domestic spending while failing to dent his five-year military buildup. They were concerned that the act of sending such an inflexible budget proposal to Capitol Hill would tarnish his leadership image while reinforcing the view that he is unfair.

Since then, the president, prodded by his staff, has embarked on a compromise course that will be consummated in a matter of days. He has made some trims in defense, authorized negotiations that now promise to bring about a bipartisan solution to rescue Social Security, agreed to look at plan for possible higher taxes in the future and signed off on a set of economic assumptions that administration officials are convinced will be unassailable.

Senate Budge Committee Chairman Pete V. Domenici (R-N.M.), a frequent critic of past administration economic assumptions, said yesterday that the White House is making "a very bona fide effort to come up with realistic solutions."

"There is no doubt the president has to take the leadership role and the budget has to be the instrument," Domenici said. "And that requires a number of realistic solutions to confront budget problems."

Whether the White House effort is aimed at the policy goal of solving these problems or the political one of shoring up a sinking president, there is no doubt that Reagan intends to make an effort in the days ahead to demonstrate that he can exercise convincing leadership in the difficult third year of his presidency.