An irreducible character of a finite group G is called quasi p-Steinberg character for a prime p if it takes a nonzero value on every p-regular element of G. In this paper, we classify the quasi p-Steinberg characters of Symmetric (Sn) and Alternating (An) groups and their double covers. In particular, an existence of a nonlinear quasi p-Steinberg character of Sn implies n ≤ 8 and of An implies n ≤ 9.