Building on proposals made by the Republican senator Phil Gramm, and the veteran Democrat Edward Kennedy, they agreed that the evidence against Mr Clinton should be presented first, and only then should the question of whether to call witnesses be decided. In the meantime, a bipartisan committee would consider how witnesses would be treated if they were called.
The framework of a solution emerged 24 hours after the opening of only the second presidential impeachment trial in US history, after the Senate had spent the morning in a rare informal session in an attempt to break days of stalemate over how the trial should proceed.
It was adjourned on Thursday afternoon, immediately after the senators had been ceremonially sworn in as jurors, with no date or timetable agreed for its resumption.
Without a resolution, the United States was in the unique - and constitutionally unsettling - position of having the President theoretically on trial for his job, with no certainty about the terms of the trial or its duration. In almost any other country such a situation would be regarded as a major constitutional crisis.
With public interest and concern about the trial apparently minimal, however, and a congressional tradition of negotiating and deal-making on the small print of legislation, the view had been that a compromise would be found.
The big sticking-point was whether the Senate should call "live" witnesses. When the House of Representatives conducted its hearings, it called only lawyers: the independent prosecutor Kenneth Starr, who conducted the investigation into the Monica Lewinsky affair, and lawyers for Mr Clinton.
They took "as read" the evidence and transcripts of testimony contained in the Starr Report and supporting documents. The Senate was polarised on whether to follow the House example or not. And, while the compromise reached yesterday should allow the trial proper to begin, the underlying problem may only have been postponed.
On one side are a majority of Democrats (effectively toeing the line of the White House) who oppose calling witnesses. Their most frequent argument is that it could drag out proceedings for months and impede the workings of government.
Behind this, however, lies another reason: the fear that Ms Lewinsky, Linda Tripp, Vernon Jordan or the President's private secretary might say things under cross-examination - perhaps inadvertently - that damaged the President's case. Witnesses would introduce an element of unpredictability that the White House desperately does not want.
On the other side are Republicans, probably a majority of them, who say that the Constitution requires a trial, and a trial should entail consideration of all the evidence and questioning of "live" witnesses. Senators, they believe, should judge the merits of the case for themselves. Hardliners also believe there may be more stones to be upturned.
One reason why Democrats, who are in the minority, were able to block an agreement so effectively and provided the basis for yesterday's compromise was that some Republicans also worry about the risks presented by witnesses. On the right are those who do not want their constituents to see them in a forum that might include Ms Lewinsky discussing the more intimate details of her anatomical contact with the President. On the left are those who fear that their more "moderate" constituents would blame them for forcing a popular president to trial, and vote them out at the next election.
The Senate Republican Majority Leader, Trent Lott, said that under the emerging plan the trial would begin next week with opening presentations. Meanwhile, four senators, two from each party, would begin working on the issue of calling witnesses.
"There's a general outline agreement. They're going to work the details out," said the Republican Jim Bunning after all senators had an extraordinary private meeting in the Old Senate Chamber. John Breaux, a Democrat, said that "ultimately there will be a vote on witnesses", indicating that it would be up to the full Senate to decide the issue.
Mr Lott said that under the emerging plan there was "no preclusion of witnesses and no inclusion of witnesses ... When we get to that point, we will make the decision then."
The Senate Democratic Leader, Tom Daschle, expected an "overwhelming, bipartisan" vote in favour of the package once it was drafted and submitted to the Senate.