  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 article, we classify the quasi $p$-Steinberg characters
of Symmetric ($S_n$) and Alternating ($A_n$) groups and their double covers. In
particular, an existence of a non-linear quasi $p$-Steinberg character of $S_n$
implies $n \leq 8$ and of $A_n$ implies $n \leq 9$.
