Special counsel Robert Mueller departing after briefing members of the Senate last month on his investigation into potential collusion between Russia and the Trump campaign. PHOTO: JOSHUA ROBERTS/REUTERS

WASHINGTON—In 1974, some of President Richard Nixon’s lawyers advised the president that he could pardon himself. In 1992, the Iran-Contra special prosecutor reached the opposite conclusion regarding President George H.W. Bush.

Neither president took that step, and constitutional scholars say the question of the presidential self-pardon remains unresolved.

But the U.S. Constitution, unlike many state governments, centralizes prosecutorial authority under the president. That means he could forbid the Justice Department from investigating or pursuing criminal charges against anyone, including himself, so he may never reach the point of having to pardon himself.

Trump officials say the issue won’t come up because there was no wrongdoing.

“I’m not sure if he has the right to [pardon] himself or not,” White House communications director Anthony Scaramucci said Sunday on CNN. “But it doesn’t matter, anyway, because that is another one of those stupid hypotheticals. He’s not going to have to pardon himself, because he’s done absolutely nothing wrong.”

The president himself tweeted Saturday, “While all agree the U. S. President has the complete power to pardon, why think of that when only crime so far is LEAKS against us.FAKE NEWS.”

The Justice Department has taken the position that a president can’t be prosecuted. Past special prosecutors have disagreed.

“It is very likely that a president is subject to federal indictment. No one is above the law in this country,” says Ronald Rotunda, a law professor at Chapman University in Orange, Calif., who worked both for the Senate Watergate Committee and later Kenneth Starr, the independent counsel whose investigation of President Bill Clinton led to impeachment and acquittal.

Last week, the New York Times reported on a memo obtained under the Freedom of Information Act in which Mr. Rotunda advised Mr. Starr that the president was subject to indictment.

“I have thought about these issues for years, beginning with my work on the Watergate Committee,” Mr. Rotunda said. “Starr’s request for a legal opinion forced me to think about it more carefully and see what is most likely the law.”

But past special prosecutors such as Mr. Starr operated under broad statutory authority that since has expired. Under current law, special counsel Robert Mueller, who is investigating alleged Russian interference in the U.S. presidential election, only has the authority to recommend to higher-ups that indictments be sought, says Frank Bowman, a law professor at the University of Missouri who publishes the ImpeachableOffenses.net blog.

Even if Justice Department attorneys obtained an indictment before the president issued an order canceling the investigation—or defied such an order—Mr. Trump could replace hostile officials with those willing to follow his orders. And if a “runaway” federal grand jury voted for an indictment on its own, a prosecution couldn’t proceed without a signature from a Justice Department lawyer, Mr. Bowman says.

“The judge has no independent power to create a prosecutorial authority,” he says.

A president attempting to derail an investigation could wreak havoc with traditional concepts of law and order, legal experts say. Because he has the power to pardon anyone for a federal crime, with the possible exception of himself, a president could in theory pardon individuals for obstructing an investigation if the offense took place under federal jurisdiction—as is all of Washington, D.C.

But there’s a far easier course if Mr. Trump acts on his assertion that the investigation against him is an illegitimate “witch hunt”: He could fire Mr. Mueller.

“If Trump takes Mueller off the board, he can pretty much stop the whole thing,” Mr. Bowman says. At that point, the inquiry’s only avenue “probably rests on the midterm elections of 2018”—and whether the Democrats can seize a chamber of Congress, and with it the ability to pursue their own investigations.