A bounded linear Hilbert space operator T is said to be k-quasi-class A operator if it satisfy the operator inequality $T^{*k}{mid}T^2{mid}T^k{geq}T^{*k}{mid}T{mid}^2T^k$ for a non-negative integer k. It is proved that if T is a k-quasi-class A contraction, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator $D=T^{*k}({mid}T^2{mid}-{mid}T{mid}^2)T^k$ is strongly stable.