Prospects for a constitutional amendment requiring a balanced Federal budget, which seemed likely to sail through Congress only a few weeks ago, dimmed dramatically today when Senator Robert C. Byrd, a veteran vote-counter and master of Senate procedure, predicted that the Senate would kill it.

As chairman of the Senate Appropriations Committee, Mr. Byrd has decisive power over the pet projects of every senator, although he said he was not applying this influence in his meetings with one senator after another to argue against the amendment.

Today Senator Byrd, a West Virginia Democrat, told reporters, "In the final analysis, Congress will not approve the amendment." 'Mischief' and 'Damage' Seen

"Once members are really informed as to the mischief this amendment could do, and the damage it could do to the country and to the Constitution," he said, "I just have faith that enough members will take a courageous stand against the amendment."

The amendment's sponsor, Senator Paul Simon, said of Mr. Byrd's remarks, "It is possible he is correct" about the amendment's prospects.

Mr. Simon, an Illinois Democrat, added: "I don't need to tell you that when the chairman of the Senate Appropriations Committee talks, senators pay attention. We all have things we want to get through appropriations."

The amendment has been under steadily increasing attack in recent days, and its prospects now appear uncertain at best.

Senator Byrd and other opponents call the amendment a smokescreen that will allow lawmakers to claim action against the deficit while still postponing hard budgetary decisions. Supporters argue that all other efforts to cut the deficit have failed and it is time for the ultimate discipline of a constitutional amendment.

Passage by the House is still considered probable when the amendment comes up there next week. But in the Senate, which will act after the House does, Mr. Byrd may be able to rally either the 34 votes required to deny the amendment the two-thirds majority required under the Constitution, or to filibuster successfully against it.

Two weeks ago he warned the Senate of a possible filibuster, saying, "Unlimited debate is one of the cornerstones on which this institution, the U.S. Senate, rests."

While it takes 41 votes to sustain a filibuster, some senators might find it easier to cast a procedural vote for unlimited debate than to vote against the popular amendment directly. Effective Lobbying Cited

Senator Simon said in an interview, "As of two weeks ago, we had the votes." But he added that Mr. Byrd's lobbying had had its effect.

Mr. Simon said Mr. Byrd would never threaten senators that their pet projects would be the first to go if the budget had to be balanced, but some senators might draw that conclusion anyhow.

An aide to Mr. Byrd insisted that when he met with other senators he assured them that his position as chairman of the Appropriations Committee was not a consideration and that they should consider his arguments strictly on their own merits.

Another measure of opposition was sounded at a news conference here by the Economic Policy Institute. The institute made public a statement of opposition signed by 447 economists, including seven who won the Nobel Prize in Economics. Making Recessions Worse

"When the private economy is in recession," they said, "a constitutional requirement that would force cuts in public or tax increases could worsen the economic downturn, causing greater loss of jobs, production and income."

The Nobel laureates signing the statement were Kenneth J. Arrow of Stanford University, Lawrence R. Klein of the University of Pennsylvania, Herbert A. Simon of Carnegie Mellon University, James Tobin of Yale University and Franco Modigliani, Paul A. Samuelson and Robert M. Solow of the Massachusetts Institute of Technology.

There are different House and Senate versions of the amendment pending, but they are similar. Senator Simon's proposal requires the President to submit a balanced budget to Congress. It compels Congress to enact and abide by annual budgets in which spending does not exceed receipts. That requirement could be suspended, a year at a time, by votes of 60 percent of both houses.

The basic argument for the amendment is that all other narrower measures to cut deficit spending have failed, and that only the compulsion of a constitutional amendment will work.

Mr. Simon said at a news conference today that the amendment "sets up a process of fiscal discipline that will mean painful choices sooner, rather than later."

The amendment, an idea that has been around for years, has gathered steam this year as Congress' own standing declined and deficits continue to increase, with the recession and the savings and loan bailout overcoming the hopes of a 1990 budget agreement between Congress and President Bush that was supposed to curb the deficit.

Senator Byrd, at the Economic Policy Institute meeting, called the proposal an "easy way out."

He and other critics have argued that the amendment itself would accomplish nothing except to give lawmakers political cover; Congress would still have to make the same difficult decisions it has so far been unable to make about what spending to cut or what taxes to increase.

Besides warning that a balanced budget requirement would make recessions worse, the economists said that it would lead to new budgetary gimmicks, invite the courts to meddle with the details of taxes, spending and the economy, and generally discourage constructive budget policies.