“I think,” he added, “when other administrations see that they say, Well that’s what leadership should be. And when faculty see that they say, That is not what university leadership should be. It’s the style of a maverick C.E.O.”

The debate over Dr. Sexton’s presidency will come to a head this week. The faculty of the university’s largest school, Arts and Science, has scheduled a five-day vote of no confidence. Given Dr. Sexton’s international stature, the vote may serve as the most important referendum yet on the direction of American higher education.

The job description for university presidents has changed significantly in recent years. In a time of shrinking resources and rising costs, leaders must, of course, raise money; N.Y.U.’s recent $3 billion campaign set a national record. But they must also raise their institutions’ profiles, forge strategic and business alliances, and plot digital strategy. Dr. Sexton, who declined to be interviewed for this article but later took part in an on-camera interview, has charged ahead on all those fronts, both in New York City and around the world.

The Global Network University that he established — with 12 international study centers in places like Accra, Ghana; Buenos Aires; and Paris, as well as a full degree-granting campus in Abu Dhabi and another one set to open next year in Shanghai — has turned N.Y.U. into a worldwide brand and broadened its students’ horizons. At a State Department reception in 2011, Hillary Rodham Clinton, then secretary of state, praised his “vision to expand his university internationally while maintaining its reputation for excellence and academic freedom.”