Nelson Cunningham, a former prosecutor, writes confidently,
There is a debate, fomented by the president’s lawyers and by conservative legal commentators, over whether Trump will submit to an interview with Mueller’s team. But the legal precedent is crystal clear: He has no choice.
Presidents going back to Thomas Jefferson have submitted to court subpoenas. Presidents Grant, Ford, Carter and both Bushes provided testimony in criminal investigations or court matters. The Supreme Court in U.S. v Nixon  unanimously held that a president could not resist a criminal trial subpoena to turn over the Watergate tapes. In 1998, the court ruled, again unanimously, in Clinton v. Jones that President Clinton could not claim immunity during his presidency from civil charges emerging from his private conduct. The court left it to the District Court to determine when and how the president could be called to testify, but there was no question that he could, and he was.
And if a president can be compelled to submit to a civil deposition and trial, then there is no question that he can be called in a criminal matter. Indeed, after Clinton v. Jones, President Clinton gave testimony not only in the civil case, but he spent almost six hours being grilled by independent counsel Starr’s team. In deference to the office and his schedule, Clinton testified from the White House, but the grand jury watched by live video feed from the courthouse. (I will pass over the irony that it was conservative lawyers who pressed for the unanimous Clinton v. Jones decision against Clinton, which now bites back against a Republican president.)
He might be responding to, among others, Theodore Olson, a former solicitor general, who wrote,
The collective significance of the Supreme Court’s opinion in [United States v. Nixon, 1974] and Chief Justice Marshall’s conclusions in the [Aaron Burr treason case in 1807] is that in a criminal case, constitutional imperatives might require a president to submit to compulsory judicial process. A defendant in a criminal case has a constitutional right to confront the evidence against him and must be able to command production of that evidence, even if it is in the hands of the president.
On the other hand, a prosecutor also has a strong claim to evidence essential to the enforcement of criminal statutes. While the president’s presumptive privilege against compulsive testimony would be entitled to considerable deference, that claim could be overcome if the prosecutor makes a strong showing that he must have specific essential evidence that is not procurable from other sources and that he is not simply engaged in a fishing expedition.
As things currently stand, Mueller has made no application to a court for a subpoena to the president based on showing a need for specific evidence essential to his prosecution and not available from other sources. News sources have published a long list of subjects said to be of interest to the special counsel, but these subjects, if they are, indeed, coming from Mueller’s office, are in many cases vague, open-ended and highly generalized—all the earmarks of casting a very wide net to see what might be snared. He’s going to have to do better than that—the usual open-ended general subpoenas won’t work.
Cunningham doesn’t fully answer Olson’s claim that a subpoena on the president would require specific evidence. (There is an apparent discrepancy in the citations for Clinton v. Jones. One says 1997; the other, 1998.)
Theodore B. Olson, “Mueller v. Trump,” Weekly Standard, May 18, 2018, https://www.weeklystandard.com/theodore-b-olson/mueller-v-trump
Nelson W. Cunningham, “3 Predictions for What Mueller Will Do Next,” Politico, May 21, 2018, https://www.politico.com/magazine/story/2018/05/21/3-predictions-for-what-mueller-will-do-next-218410