Should the investigation into the leak be carried out by the Senate? Or by an independent prosecutor? By the congressional watchdog General Accounting Office? Or by the FBI?
"The leadership is stonewalling," said Sen. Hank Brown, R-Colo., a member of the Senate Judiciary Committee and author of a resolution calling on Majority Leader George J. Mitchell, D-Maine, to appoint a special counsel.
"The Democrats want to sweep this under the rug."
Not so, said an aide to Mr. Mitchell.
"He is consulting with his colleagues over the best way to proceed," she said.
Indeed, shortly before Tuesday's confirmation vote, Mr. Mitchell took to the Senate floor to decry leaks and assure colleagues that "if I can determine the identity of the person who did it, I'm going to try to see that the person is appropriately punished."
That, however, may be easier said then done, given the elusive nature of leaks and the apparent anonymity of the leakers.
Any investigation will begin with a few tenuous leads.
Ms. Hill's allegations of sexual harassment were published first in the Oct. 6 editions of Newsday, and copies of the article by reporter Timothy Phelps moved on news wires the night before.
The morning after Newsday's account was circulated, Nina Totenberg, a National Public Radio reporter, also broadcast a version of the story, and the confirmation process of Judge Thomas was sent down a previously untraveled path.
Complicating the situation is uncertainty about whether a senator or staffer leaked Ms. Hill's statements to the reporters.
Instead, a lawmaker or aide could have provided the information to any in a constellation of interest groups opposed to the Thomas nomination.
An official of one of the outside groups, in turn, could have leaked the material to the reporters.
The principals, not surprisingly, aren't saying.
FBI reports are rarely made public -- especially in a case such as this -- because they contain largely unprocessed and always unsworn statements and allegations.
The leak of such a report, whether to a reporter or an interest group, would constitute a violation of Senate rules.
"We can't just stand by and let something like this go unpunished," said Sen. Alan K. Simpson, R-Wyo.
"That's an invitation to anarchy," he said.
In the meantime, the invitation to investigate continues to cause heartburn as lawmakers dicker over who should do the investigating.
The Senate Select Committee on Ethics could launch it, although the idea of senators investigating senators appears to have fallen into disrepute of late.
"The ethics committee? That's a joke," said Sen. Dennis DeConcini, D-Ariz., himself the recent target of an ethics inquiry.
"I know this from personal experience."
Even the committee's chairman, Sen. Howell Heflin, D-Ala., looks askance at the idea of an investigation spearheaded by his panel.
"The ethics committee? Why, that's a sieve," he said.
The GAO could launch an inquiry.
"They're honest and independent," said Sen. Paul Simon, D-Ill., who, like every other Democrat on the Judiciary Committee, has denied that he or his staff leaked anything to the press.
Mr. Simpson also issued a blanket denial on behalf of all the Republican members of the committee.
Republicans appear particularly enamored of the idea of an FBI inquiry or the appointment of a Justice Department special prosecutor.
Such an action, though, could run afoul of constitutional separation-of-powers concerns, which prevent the executive branch from overseeing legislative branch activities.