In an impeachment trial in the Senate, President Trump would look up to see one of his Washington establishment foes, Supreme Court Chief Justice John G. Roberts Jr., presiding over the historic proceedings from the dais in the upper chamber.

Mr. Trump, who has clashed with Chief Justice Roberts over the perceived political bias of the federal courts, would have to count on him for a fair hearing when the fate of his presidency hangs in the balance.

It’s a prospect that has caused rumblings in Washington that the chief justice should recuse himself.

John Cardillo, a conservative radio personality and host on Newsmax TV, sounded the recusal alarm.

“There is already a crisis of confidence among the American people that we have a fair system of justice. When you have a chief justice of the Supreme Court overtly making comments that are derogatory to the president of the United States, take all speculation out of the process,” he said.

What’s more, the recusal question extends down the bench of the high court. All of the justices likely will play roles in deciding Congress’ subpoena power over the executive branch as House Democrats conduct an impeachment inquiry.

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Chief Justice Roberts and Justice Ruth Bader Ginsburg have publicly commented on Mr. Trump in recent years. Justice Ginsburg went as far as calling him a “faker,” and the chief justice sent a sharp rebuke to the commander in chief over the political independence of the judiciary.

The issue of potential bias was raised about a year after Chief Justice Roberts released an unusual statement indirectly firing back at the president over his politicization of judges based on the presidents who appointed them.

“We do not have Obama judges or Trump judges, Bush judges or Clinton judges,” Chief Justice Roberts said in a statement in November 2018 issued from the court’s public information office.

“What we have is an extraordinary group of dedicated judges doing their level best to do equal right to those appearing before them,” he said.

The president received the rare rebuke after criticizing a ruling from a district judge in California against his administration.

“Sorry Chief Justice John Roberts, but you do indeed have ‘Obama judges,’ and they have a much different point of view than the people who are charged with the safety of our country,” the president fired back on Twitter.

Chief Justice Roberts‘ defenders insist that the eyebrow-raising statement does not demonstrate a conflict of interest or at least not one significant enough to require him to step aside from overseeing Mr. Trump’s impeachment trial.

“Roberts’ comments were about as even-keeled as they get,” said Josh Blackman, a professor at South Texas College of Law, adding in an impeachment trial, “The chief justice’s role is fairly limited.”

Mark Graber, a constitutional law professor at the University of Maryland, went a step further.

“There is nobody on the court who ought to be recused,” he said.

Still, the justices likely will play a critical role in the subpoena fight over which administration documents and testimony House Democrats can have.

Without directly weighing in on the issue of impeachment and Congress’ subpoena power, a federal court last week ruled that Mr. Trump’s accounting firm had to turn over tax documents to House Democrats in one of their many probes into Mr. Trump and his financial dealings.

The case involving the financial documents could test the president’s executive authority to thwart the congressional subpoenas.

Ilya Shapiro, who publishes the Cato Institute’s “Supreme Court Review,” said an emergency petition to halt a lower court decision in the impeachment inquiry likely would be filed with the justices in the future.

If so, the debate over recusals might be raised against Justice Ginsburg, whose comments during the 2016 campaign about then-candidate Donald Trump likely crossed the line of what is permitted by a justice.

Justice Ginsburg has apologized for calling Mr. Trump a “faker.”

“He has no consistency about him. He says whatever comes into his head at the moment. He really has an ego,” she told CNN in 2016. “How has he gotten away with not turning over his tax returns? The press seems to be very gentle with him on that.”

In response, Mr. Trump called for the senior justice’s resignation.

Mr. Graber said the liberal icon’s remarks about the 2016 campaign have been worked out and she has weighed in on several legal challenges involving the administration during the president’s first three years in office.

Justices Neil M. Gorsuch and Brett M. Kavanaugh “are Trump appointees — nobody is talking about recusing them, and for good reason,” Mr. Graber said.

Mr. Shapiro said Chief Justice Roberts‘ role in the impeachment proceeding would be more like a “traffic cop than trial judge,” and he flatly dismissed any questions about Justice Ginsburg’s alleged bias.

“Her comments in no way prejudged the legal issues at stake,” he said.

If Chief Justice Roberts did decide not to preside over a Senate impeachment trial, then the most senior justice, Justice Clarence Thomas, would step in, Mr. Blackman said.