“You can be disappointed, but don’t be surprised,” said Karen Dunn, a former prosecutor and White House lawyer under President Barack Obama who advised Mrs. Clinton during her campaign. “This is exactly what he said he would do: use taxpayer resources to pursue political rivals.”

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.”