The purpose of this paper is to prove that the noncommutative version of the Singer-Wermer Conjecture is affirmative under certain conditions. Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D:A{ ightarrow}A$ such that $D(x)^3[D(x),x]{in}rad(A)$ for all $x{in}A$. In this case, we show that $D(A){subseteq}rad(A)$.