Upholding a statute barring aliens from teaching in public schools, the [Supreme] Court reasoned that the "distinction between citizens and aliens, though ordinarily irrelevant to private activity, is fundamental to the definition and  government of a State. . . . It is because of this special significance of citizenship that governmental entities, when exercising the functions of government, have wider latitude in limiting the participation [**6] of noncitizens." Ambach, 441 U.S. at 75 (emphasis added).
So while constitutional, what is the extent of this prohibition? I think a good number of academics in higher US education are (at least initially) non-US-citizens. And even in lower-level US public schools there seem to be a number of foreigners.
More than 3,000 teachers given visas
The U.S. State Department says 3,252 foreign teachers were given exchange visas in 2018. The countries sending the most teachers were the Philippines, Spain, Jamaica, China and France.