In this paper, a new normalized integral operator is introduced, defined by an integral containing a power of the classical Bessel function of the first kind of order $\nu \in \mathbb{N}$ multiplied by $t^{\gamma -1}$. Sufficient conditions for this operator to be starlike and convex in the open unit disk are determined. To establish these geometric criteria, we apply the theory of differential subordination, initiated by S. S. Miller and P. T. Mocanu, also known as the method of admissible functions.