In December, Senator Ben Sasse, Republican of Nebraska and a member of the Senate Judiciary Committee, called on the Justice Department to investigate the actions of Mr. Acosta and other federal officials. But no Republican in either chamber has called for hearings, and it remains unlikely that the Republican-controlled Senate would take up a potentially embarrassing episode involving a respected lawyer who was, until recently, one of Mr. Trump’s least controversial cabinet picks.

But Democrats said the matter was ripe for congressional involvement. They suggested that Mr. Acosta was not completely forthcoming when he played down his role in the Epstein negotiations during his March 2017 confirmation hearings. A November 2018 story by The Miami Herald documented Mr. Acosta’s extensive interactions with Mr. Epstein’s lawyers.

“What makes me maddest is that the plea deal Acosta agreed to was about prostitution,” said Senator Tim Kaine, a Virginia Democrat who grilled Mr. Acosta on the Epstein case during his confirmation hearings.

Under an unusual “nonprosecution” deal negotiated by Mr. Acosta, then the United States attorney for the Southern District of Florida, the government agreed to drop the investigation in exchange for Mr. Epstein’s confession to prostitution and an 18-month sentence, of which Mr. Epstein served 13 months.

Last week, United States District Judge Kenneth Marra ruled that Mr. Acosta and his team of prosecutors violated the Crime Victims’ Rights Act because they failed to inform the victims in the case about the details of the plea deal. The Justice Department’s professional conduct division is investigating whether any rules or laws were broken.

On Friday the White House press secretary, Sarah Huckabee Sanders, said the administration was “looking into” Mr. Acosta’s role in securing the plea deal.

Mr. Acosta may not find the loyalty in conservative ranks that other troubled cabinet secretaries have found. Among lobbyists and employment lawyers representing business, there has been frustration with the speed at which Mr. Acosta has moved to tilt employment policy in a more business-friendly direction.