Robert Mueller should testify publicly about the special counsel’s investigation into the 2016 election, former independent counsel Kenneth Starr said in an interview aired Friday.

Mr. Starr, who led the investigation that resulted in impeachment proceedings being initiated against former President Bill Clinton, made the remark during an appearance on the FOX Business Network, citing “unanswered questions” remaining after Mr. Mueller recently completed the special counsel’s probe.

“We need to take a careful look at this, and I’m pleased that there’s a sense on both sides of the aisle,” said Mr. Starr, a lawyer and Fox News contributor.

“I know the president would like this to just go away, but I think there’s a sense on both sides of the aisle that we need to hear from Bob Mueller,” he said.

A former FBI director, Mr. Mueller was appointed special counsel in May 2017 and tasked with investigating Russian interference in the 2016 U.S. elections and related matters, including possible obstruction of justice on the part of President Trump. He released his findings in a report released April 18, and he announced his resignation as special counsel on May 29.

Mr. Mueller said that 448-page report speaks for itself and has resisted calls to discuss his findings before Congress: Russians interfered in the race in a “sweeping and systematic fashion,” while Department of Justice policy precluded prosecutors from criminally charging a sitting president, he said.

Fifty-six percent of registered voters want Mr. Mueller to publicly testify about the special counsel’s probe, according to polling released last month.

House Judiciary Committee Chairman Jerry Nadler said Wednesday that he’s “confident” Mr. Mueller will oblige, meanwhile.

“We want him to testify openly. I think the American people need that. Frankly, I think that’s his duty to the American people,” said Mr. Nadler, New York Democrat. “We’ll make that happen.”

Mr. Starr, 72, was appointed independent counsel in 1994 and put in charge of probing the “Whitewater” real estate scandal dogging the Clinton administration at the time. His results, published in 1997 in the so-called “Starr Report,” revealed that Mr. Clinton had lied about a sexual relationship with a former intern while in office. Republicans in the House of Representatives subsequently voted to impeach the president, but the GOP-controlled Senate failed to secure the two-thirds voted needed to remove him from the White House.