In oral arguments, Gilbert Davis, lead attorney for Ms Jones, urged that the case go forward forthwith, as ordered last year in a federal appeals court judgment Mr Clinton is now appealing. "Justice delayed is justice denied," Mr Davis said, noting that as a sitting president, Mr Clinton had already testified at length in two separate Whitewater trials, dealing with events before he entered the White House.
But in pleading for a postponement, Robert Bennett, representing the President, stressed the overriding importance of not disrupting the executive governance of the country. "Here you have a sitting President. We are saying 'We'll give Ms Jones her day in court. Only let's not do it now'."
Mr Bennett seemed on shakier ground when he cited a 1982 court ruling for Richard Nixon, in effect granting a president immunity from civil suits arising from his conduct in office. The comparison seemed irrelevant, Chief Justice William Rehnquist and various colleagues complained, noting that the alleged incident dated back to 1991, and had nothing to do with Mr Clinton's duties as Governor of Arkansas.
Justice Sandra Day O'Connor wondered whether the temporary immunity sought by Mr Bennett would apply if a case involving the President concerned a domestic issue such as child custody. Antonin Scalia, the leader of the court's conservative faction, was sceptical about Mr Clinton's contention that the case would be a time-consuming distraction. It might mean the President had to give up some golf games, he noted.
Ms Jones's lawyers said the case was relatively uncomplicated, a matter of one person's word against another's. "This will not take very long," Mr Davis told reporters after the hour of oral arguments. "A president is the same as you or me in his personal capacity ... his office has privileges and immunities, but he cannot claim a personal privilege."
That is the issue the justices must rule on, before July. However, another lawyer for Ms Jones again left open the possibility of an out-of-court settlement, perhaps similar to the one the two sides almost achieved in 1994, just before Ms Jones filed her $700,000 (pounds 417,000) suit that Mr Clinton had made sexual advances to her in a Little Rock hotel.
Many court-watchers suspect they will "split the difference", deferring the trial until Mr Clinton has left office but allowing gathering of evidence, or "discovery," to go ahead.
Ms Jones's representatives again denied she is a gold-digger, or the creature of right-wing anti-Clinton groups. She wanted simply to restore her good name. "Money is not a factor in this case," Mr Davis said.
And that seemed the prevailing view of demonstrators outside the Supreme Court building. Many bore placards with slogans like "President or King?" and "Stop Ducking Your responsibility, Mr President." A president "has the right to protection in the course of his duties," said one protester, Washington office worker Kristi Hamrick, "but sexual harassment is not one of those duties."