Let $sigma$ be an endomorphism and I a $sigma$-ideal of a ring R. Pearson and Stephenson called I a $sigma$-semiprime ideal if whenever A is an ideal of R and m is an integer such that $A{sigma}^t(A);{subseteq};I$ for all $t;{geq};m$, then $A;{subseteq};I$, where $sigma$ is an automorphism, and Hong et al. called I a $sigma$-rigid ideal if $a{sigma}(a);{in};I$ implies a $a;{in};I$ for $a;{in};R$. Notice that R is called a $sigma$-semiprime ring (resp., a $sigma$-rigid ring) if the zero ideal of R is a $sigma$-semiprime ideal (resp., a $sigma$-rigid ideal). Every $sigma$-rigid ideal is a $sigma$-semiprime ideal for an automorphism $sigma$, but the converse does not hold, in general. We, in this paper, introduce the quasi $sigma$-rigidness of ideals and rings for an automorphism $sigma$ which is in between the $sigma$-rigidness and the $sigma$-semiprimeness, and study their related properties. A number of connections between the quasi $sigma$-rigidness of a ring R and one of the Ore extension $R[x;;{sigma},;{delta}]$ of R are also investigated. In particular, R is a (principally) quasi-Baer ring if and only if $R[x;;{sigma},;{delta}]$ is a (principally) quasi-Baer ring, when R is a quasi $sigma$-rigid ring.