The office of the independent prosecutor, Kenneth Starr, meanwhile, let it be known that his investigations into the President were far from over and that additional damaging documents were to come.
The White House was said to be seriously considering a suggestion that Mr Clinton go in person to Capitol Hill to testify to the House judiciary committee - the committee that had threatened his presidency by voting to release the tape of his testimony, the committee that must consider the evidence for impeachment. Whether this would forestall moves towards impeachment hearings, however, was doubted.
The proposal that Mr Clinton - whose relations even with Congressional Democrats have never been warm - should appear before the judiciary committee had first been made by Senator John Kerry, a Massachusetts Democrat, shortly before the videotape was aired. But it was alluded to repeatedly, by both Democrats and Republicans, yesterday as the committee reconvened.
Among fervent Clinton supporters, however, there was a feeling that any appearance by the President before the judiciary committee should be agreed only as part of a deal that would end Mr Starr's investigation, and avert the threat of impeachment proceedings.
The former Senator and presidential candidate, Robert Dole, told a cable television phone-in programme that he had heard the White House was planning some "bold move". "Obviously the President's reaching out, he'd like to find some way to end this," Mr Dole said, but he doubted whether this, by itself, would resolve Mr Clinton's difficulties. He predicted that Congress will move ahead, "maybe some time next week or the following week and vote on whether to proceed with an impeachment inquiry".
That vote could come as early as next week. According to Mr Dole, any "deal" would be most likely to be agreed between then and February, when a new Congress reconvenes after the mid-term elections.
The retiring Senator and constitutional historian, Daniel Pat Moynihan, has said Congress could dispatch the whole process - from the decision on an impeachment inquiry through to a vote in the Senate judiciary committee - within six weeks, but there are few who believe this timetable to be feasible or likely.
The shift of attention from the precariousness of the President's position to the possibility of a deal was one clear result of Mr Clinton's broadcast testimony and the flood of more details about his relationship with the former White House trainee, Monica Lewinsky.
Almost immediately the broadcast was over on Monday afternoon, a perceptible shift could be observed in the tenor of "insider" comment from the big political arena to the small print of judicial process.
The criticism that had questioned Mr Clinton's credibility and argued that he could not remain in office was replaced by debate about whether there was sufficient evidence to support charges of perjury and abuse of power, as set out by Mr Starr in his report, which also said that Mr Clinton lied under oath "three times" during the grand jury testimony.Reuse content