Washington — President Reagan's newest arms control initiative, unveiled in his speech at Eureka College in Illinois, attempts to follow a difficult trajectory through the concerns of at least four crucial audiences:
* The American public. Recent polls show Americans supporting the idea of a nuclear weapons freeze by 3 to 1. Nearly half the public thinks Reagan has not done enough to limit the buildup of such weapons.
* The US Congress. Both the Senate and the House are in the midst of nuclear arms limitation hearings. More than 40 resolutions are under consideration in the Senate Foreign Relations and House Foreign Affairs committees. Subject matter ranges from what to do about SALT II, to the nuclear freeze vs. nuclear reduction debate. US legislators have been trying to find a focus for arms control deliberations, without a firm Reagan proposal to steer by.
* Western Europe's leaders. The NATO chiefs of state are anticipating a meeting with the President in early June. The European nuclear freeze movement is seeking to build a bridge to its American counterpart, with a prospect of large demonstrations. In addition, the US and its allies have a host of economic and defense issues to consider, in which the question of US-Soviet nuclear detente plays a significant role.
* The Kremlin. The Soviets rejected a 1977 and '79 proposals by Jimmy Carter to reduce, rather than limit, nuclear arsenals. Reagan's arms talk advisers claim the Republican's proposal has a better chance of success because ''it will not come as a major surprise'' to the Soviets as Carter's apparently did.
To get in front of building domestic pressure for movement toward arms control, President Reagan felt compelled to move even more quickly than he and many of his advisers would have wished toward concrete arms talks proposals.
In a first phase of negotiation, Reagan wants the United States and the Soviet Union to reduce their existing arsenals by a third -- from about 7,500 deployed nuclear warheads each to 5,000. In a second phase, he wants both nuclear antagonists to reduce and rearrange their nuclear arsenals to reach ''equal throw-weight'' levels.
In the interim, explains a senior administration official, the superpowers would continue to tacitly abide by the Salt II guidelines.
With these proposals, the President has at last moved to structure, if not end, the strategic nuclear arms controversy.
''We will negotiate seriously, in good faith, and carefully consider all proposals made by the Soviet Union,'' Reagan told his Eureka audience in Illinois.
Alluding to his separate bid to meet face-to-face soon with the Soviet Union leader, Reagan added: ''And when we sit down I will tell President Brezhnev that the United States is ready to build a new understanding. . . . I will ask President Brezhnev why our two nations cannot practice mutual restraint. Why can't our peoples enjoy the benefits that flow from real cooperation? Why can't we reduce the number of horrendous weapons?''
''Everyone's been waiting for his cue,'' says Robert J. Pranger, director of foreign and defense policy studies for the American Enterprise Institute. ''He hasn't time for choice anymore. The action is with him.''
At the moment, the President suffers from ''a very bad image as a nuclear warrior,'' Pranger says. ''This whole defense buildup issue has spilled over onto the nuclear side, even though the Reagan defense budget carries on Carter's nuclear programs.''
''He has the Europeans, first of all, to address,'' Pranger says. ''They've been expecting something realistic in negotiating terms, that will pass the arms control community's inspection. By the time he gets to Bonn (June 10) he's going to be looking at protests that will try to establish some linkage between the American and European freeze movements. He's going to be looking at some Americans - prominent Americans, possibly - in Bonn. It's important to head this off.''
Reagan aides portrayed the President's Eureka address as his scene-setter for his June 2-11 European visit.