Democrats vividly recall Mr. Trump on the campaign trail vowing to prosecute Mrs. Clinton if he won. “It was alarming enough to chant ‘lock her up’ at a campaign rally,” said Brian Fallon, who was Mrs. Clinton’s campaign spokesman. “It is another thing entirely to try to weaponize the Justice Department in order to actually carry it out.”

But conservatives said Mrs. Clinton should not be immune from scrutiny as a special counsel, Robert S. Mueller III, investigates Russia’s interference in last year’s election and any connections to Mr. Trump’s campaign. They argued, for example, that Mrs. Clinton was the one doing Russia’s bidding in the form of a uranium deal approved when she was secretary of state.

Peter Schweizer, whose best-selling book “Clinton Cash” raised the uranium issue in 2015, said a special counsel would be the best way to address this matter because it would actually remove it from politics. “It offers greater independence from any political pressures and provides the necessary tools to hopefully get to the bottom of what happened and why it happened,” said Mr. Schweizer, whose nonprofit organization was co-founded by Stephen K. Bannon, Mr. Trump’s former chief strategist.

At Tuesday’s hearing before the House Judiciary Committee, Mr. Sessions denied that he was responding to Mr. Trump’s public pressure. “A president cannot improperly influence an investigation,” he said, “and I have not been improperly influenced and would not be improperly influenced. The president speaks his mind. He’s a bold and direct about what he says, but people elected him. But we do our duty every day based on the law and facts.”

Even as he rebuffed Democrats suggesting he had been compromised, Mr. Sessions pushed back against Republicans who pressed him on why he had not already appointed a special counsel. “What’s it going to take to get a special counsel?” demanded Representative Jim Jordan of Ohio.

“It would take a factual basis that meets the standards of the appointment of a special counsel,” Mr. Sessions said.

Mr. Jordan raised questions about a dossier of salacious assertions about Mr. Trump prepared last year by a firm paid by Mrs. Clinton’s campaign and the Democratic National Committee. Mr. Jordan said “it sure looks like” the Democrats collaborated with the F.B.I. to use the dossier to persuade a secret intelligence court to issue a warrant to spy on Americans associated with Mr. Trump’s campaign. “That’s what it looks like,” Mr. Jordan said.

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Mr. Sessions bridled at that. “I would say ‘looks like’ is not enough basis to appoint a special counsel,” he retorted.

Among the issues being examined, according to a Justice Department letter to the committee, is the uranium case. In 2010, Russia’s atomic energy agency acquired Uranium One, a Canadian company that at the time controlled 20 percent of American uranium extraction capacity. The purchase was approved by a government committee that included representatives of nine agencies, including the State Department.

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Donors related to Uranium One and another company it acquired contributed millions of dollars to the Clinton Foundation, and former President Bill Clinton received $500,000 from a Russian bank for a speech. But there is no evidence that Mrs. Clinton participated in the government approval of the deal, and her aides have noted that other agencies signed off on it. The company’s actual share of American uranium production has been 2 percent; the real benefit for Russia was securing far greater supplies of uranium from Kazakhstan.

Other issues mentioned in the Justice Department letter include Mrs. Clinton’s use of a private email server, which was investigated by the F.B.I. until the bureau’s director at the time, James B. Comey, declared last year that no prosecutor would press charges based on the evidence. The letter said the department was also examining Mr. Comey for leaking details of his conversations with Mr. Trump after the president fired him.

To the extent that there may be legitimate questions about Mrs. Clinton or Mr. Comey, however, the credibility of any investigation presumably would be called into question should one be authorized by Mr. Sessions or his deputy, Rod J. Rosenstein, because of the way it came about under pressure from Mr. Trump.

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Of 10 former attorneys general contacted Tuesday, only one responded to a question about what they would do in Mr. Sessions’s situation.

“There is nothing inherently wrong about a president calling for an investigation,” said William P. Barr, who ran the Justice Department under President George Bush. “Although an investigation shouldn’t be launched just because a president wants it, the ultimate question is whether the matter warrants investigation.”

Mr. Barr said he sees more basis for investigating the uranium deal than any supposed collusion between Mr. Trump and Russia. “To the extent it is not pursuing these matters, the department is abdicating its responsibility,” he said.

Mr. Trump promised last year that if elected, he would instruct his attorney general to appoint a special prosecutor to investigate Mrs. Clinton. But he backed off that shortly after the election, saying, “I don’t want to hurt the Clintons.”

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By last summer, with Mr. Mueller’s investigation bearing down, he had changed his mind. To Mr. Trump, the inquiry was a “witch hunt” based on a “hoax” perpetrated by Democrats. It was all the more galling, advisers said, because Mrs. Clinton had not been prosecuted, a frustration exacerbated by recent reports about how her campaign helped finance the salacious dossier.

“So why aren’t the Committees and investigators, and of course our beleaguered A.G., looking into Crooked Hillarys crimes & Russia relations?” he wrote on Twitter in July.

“There is so much GUILT by Democrats/Clinton, and now the facts are pouring out,” he wrote in October. “DO SOMETHING!”

“At some point the Justice Department, and the FBI, must do what is right and proper,” Mr. Trump wrote again in November. He added: “Everybody is asking why the Justice Department (and FBI) isn’t looking into all of the dishonesty going on with Crooked Hillary & the Dems.”

Mr. Trump has expressed frustration that he does not control the F.B.I. or the Justice Department. By his own account, he fired Mr. Comey while bristling at the Russia investigation that the F.B.I. director was then leading. He has also expressed anger at Mr. Sessions for recusing himself from overseeing that investigation, resulting in Mr. Mueller’s appointment, and he has refused to rule out firing the attorney general.

With his job potentially on the line, Mr. Sessions has been put in the difficult position of absorbing his president’s ire while safeguarding the department’s traditional independence. By asking prosecutors to evaluate the evidence, he has a ready-made reason not to appoint a special counsel if they do not recommend one.

“I have no idea what will happen, but this letter is entirely consistent with the A.G. later saying, ‘We followed normal process to look in to it and found nothing,’” said Jack L. Goldsmith, a former Justice Department official under the younger Mr. Bush. “The letter does not tip off or hint one way or another what the A.G.’s decision will be.”

At least one active Twitter user will be waiting for that decision.