Let $t$ be indeterminate and let $\rho: B_n \rightarrow GL_{m}(\mathbb{Z}[t^{\pm 1}])$ be a $k$-local representation of the braid group $B_n$, where $k=m-n+2$. We consider two types of extensions of $\rho$ to the singular braid monoid $SM_n$. The first type is called the $k$-local extension, which is a new concept defined in this paper. The second type is called the $\Phi$-type extension, which is given by Bardakov, Chbili, and Kozlovskaya in \cite{13}. In order to obtain a relation between these two types of extensions, we consider two homogeneous $2$-local representations of $B_n$, namely $\rho_B: B_n \rightarrow GL_{n}(\mathbb{Z}[t^{\pm 1}])$ and $\rho_S: B_n \rightarrow GL_{n}(\mathbb{Z}[t^{\pm 1}])$, and a homogeneous $3$-local representation of $B_n$, namely $\rho_F: B_n \rightarrow GL_{n+1}(\mathbb{Z}[t^{\pm 1}])$. We study the relation between the two types of extensions to $SM_n$ of these three representations of $B_n$. In fact, we prove that every homogeneous $2$-local extension of $\rho_B$ is also a $\Phi$-type extension for all $n\geq 2$; which is not the case for $\rho_S$. Also, we prove that every homogeneous $3$-local extension of $\rho_F$ is a $\Phi$-type extension for all $n\geq 3$. In addition, we study, in the case $n=2$, the faithfulness of the complex specialization of all $2$-local extensions of $\rho_B$ and $\rho_S$.