Such an appointment could give President Trump and Republicans a political counterweight to the ongoing work of special counsel Robert S. Mueller III, who is probing whether any Trump associates coordinated with the Russian government to interfere in last year's presidential election.
For that reason, Sessions's suggestion has raised fresh questions about the independence of the Justice Department in the Trump administration.
"To have the winning side exploring the possibility of prosecuting the losing side in an election — it's un-American, and it's grotesque," said John Danforth, a former special counsel who investigated the FBI's role in a violent standoff with a cult in Waco, Tex. "The proliferation of special counsels in a political setting is very, very bad."
Peter R. Zeidenberg, who once served as deputy special counsel in the probe of former White House aide Lewis "Scooter'' Libby, said "the best-case scenario" is that the attorney general is trying simply to mollify an angry president and doesn't really plan to name a special counsel.
If one is appointed to probe Clinton matters, "I think the vast majority of people at DOJ would be completely disgusted and demoralized by it,'' said Zeidenberg, referring to the Justice Department. "They don't like feeling that they are political tools to be used by the president.''
A federal law enforcement official, who spoke on the condition of anonymity to comment freely, said the move is particularly frustrating because many of the subjects Sessions says could fall under the purview of a special counsel were previously investigated by the FBI or are under investigation by the Justice Department's inspector general.
"Going back to look at issues we've already looked at — and some issues we're still looking at — what is that if not politics?" asked the official.
Sessions insisted in his testimony that he is not leaning one way or the other on the appointment of a new special counsel and that he is awaiting a detailed presentation from senior officials at the Justice Department.
"I have no prejudgment. I did not mean to suggest I was taking sides one way or the other on that subject,'' he said.
The first year of the Trump administration has been a turbulent one for the Justice Department. FBI Director James B. Comey was fired. The president has repeatedly and publicly attacked the department, the FBI and senior officials at those agencies, complaining that he wishes he could have more control over how they pursue criminal cases.
In several angry tweets Nov. 3, the president called again for investigations of Clinton and Democrats, saying, "at some point the Justice Department, and the FBI, must do what is right and proper. The American public deserves it!''
Current and former Justice Department officials could not recall the last time the department publicly dangled the possibility of a special counsel appointment. When House Republicans demanded a special counsel probe of the IRS's treatment of conservative groups during the Obama administration, the Justice Department stayed largely silent on the call until it issued a lengthy letter explaining the reasons prosecutors would not pursue charges in the case.
Also unclear Tuesday was how Sessions might appoint such a special counsel for some of the issues, given that he has recused himself from investigative matters involving the 2016 campaign due to his vocal support of Trump and fierce public criticism of Clinton.
One of the areas that Republicans want investigated is whether donations to the Clinton Foundation may have been tied to political favors by Clinton or any of her subordinates when she was secretary of state. The FBI probed that issue in 2015 and 2016, but Justice Department officials decided there wasn't enough evidence to pursue a full-fledged investigation. A separate Clinton Foundation-related federal probe has been ongoing in Virginia, but its final disposition is still unclear, according to people familiar with the matter.
In his testimony, Sessions did say that he believed his recusal would not bar him from appointing a special counsel to investigate possible wrongdoing involving Uranium One, a Canadian mining company with U.S. assets that was purchased by Russia's nuclear energy agency. Republicans have suggested that the deal, which required U.S. approval from a multiagency board that included Clinton's State Department, should have been blocked because of a separate corruption probe involving Russian uranium shipments.
Yet the attorney general also offered a mild rebuttal to suggestions by some of his fellow Republicans that there are clear connections in the uranium case that need further investigation, noting that the approval of the deal was two or three years removed from the key events in the corruption probe.
Some conservatives say a special counsel appointment could have consequences far beyond Clinton. Fox News host Sean Hannity, who is also an informal adviser to the president, has called for a special counsel to probe the uranium deal and has said that because Mueller was the director of the FBI at the time Uranium One was purchased, Mueller should be removed from his role probing the Trump White House.
"There's no way the American people can trust Robert Mueller to investigate anything Russia-related," Hannity has said.
Former federal prosecutor Andrew McCarthy said he doubted there was any prosecutor appointment or assignment that would lead to Mueller stepping down.
"It seems they're frustrated by the Mueller investigation, and they think, 'Why don't we get one to look at the Clinton stuff?' And that's just not the way it's supposed to work," he said.
McCarthy also said he thought it was a mistake, legally and politically, for the Justice Department to publicly suggest that it might appoint a special counsel.
"I don't know what Sessions is going to do here, but I don't see any good reason for the Justice Department, in a situation where it's not the administration investigating itself, to appoint a special counsel," he said. "You should never float the idea on your side, because when you float the idea, you are at least suggesting there's a possibility you may not be capable of doing the job yourself for ethical reasons."