We introduce and study two subclasses $\Omega_{\lbrack\alpha_{1}]}(A,B,\lambda)$ and $\Omega_{\lbrack\alpha_{1}]}^{+}(A,B,\lambda)$ of meromorphic $p$-valent functions defined by certain linear operator involving the generalized hypergeometric function. The main object is to investigate the various important properties and characteristics of these subclasses of meromorphically multivalent functions. We extend the familiar concept of neighborhoods of analytic functions to these subclasses. We also derive many interesting results for the Hadamard products of functions belonging to the class $\Omega_{\lbrack\alpha_{1}]}^{+}(\alpha,\beta,\gamma,\lambda)$.