Let $R$ be a Noetherian local ring, $I$ and $J$ are ideals of $R$, and $M$ and $N$ are $R$-modules. We study the relationship between the Bass numbers of $\Hom_R(N, M)$ and $H^{i}_{I,J}(N, M)$. As a consequence, it follows that if one of the following holds: \begin{itemize} \item[(a)] $I$ is a principal ideal of $R$, \item[(b)] $\dim_R M=1$, \item[(c)] $\dim_R M/{JM}=1$ (when $R$ is local and $M$ is finitely generated), \item[(d)] $\dim_R R/{J}=1$ (when $R$ is local), \item[(e)] $\dim R=1$ (when $R$ is local). \end{itemize} then $\mu^j(\frak p, H^{i}_{I,J}(N, M))$ is finite for all $i\in\Bbb N_0$ and $j\in\Bbb N_0$, whenever $N$ is finitely generated and flat, $M$ is minimax, and $\frak p\in {\rm W}(I, J)$.
