I’m going to trace back ἦν and εἶναι to their Proto-Indo-European root. I know the Greek verb εἶναι is derived from the Indo-European root *h1es-. *h1es- means ‘to be’ and φύω, from which physics and physical are derived, is derived from the Indo-European root bhū. The root bhū probably meant ‘to grow’, but also ‘to become’.
The root > as> , which still lives in our > he is> , is a very old root: it existed in its abstract sense previous to the Aryan separation. Nevertheless we know that > as> , before it could mean to be, meant to breathe.
The simplest derivation of > as > to breathe, was > as-u> , in Sanskrit, breath; and from it probably > asu-ra> , those who breathe, who live, who are, and at last, the oldest name for the living gods, the Vedic > Asura> .When this root > as> , to breathe, was felt to be inconvenient to trees and other things which clearly do not breathe, a second root was taken, > bhū> , meaning originally to grow, the Greek φυ-ω, which still lives in our own > to be> . It was applicable, not to the animal world only, but also to the vegetable world, to everything growing, and the earth itself was called Bhūs, the growing one.
F. Max Muller, Origin and Growth of Religion, pp. 191-192