The original Maxwell's equations are given by \begin{align} \nabla\cdot\textbf{E}&=\frac{\rho}{\epsilon_0}\\ \nabla\cdot\textbf{B}&=0\\ \nabla\times\textbf{E}&=-\frac{\partial\textbf{B}}{\partial t}\\ \nabla\times\textbf{B}&=\mu_0\textbf{J}+\mu_0\epsilon_0\frac{\partial\textbf{E}}{\partial t} \end{align}
Since $\nabla\cdot\textbf{B}=0$, we can write $\textbf{B}=\nabla\times\textbf{A}$. Then we can write \begin{align} \textbf{E}=-\nabla\phi-\frac{\partial\textbf{A}}{\partial t} \end{align} And we call $(\phi/c,\textbf{A})$ electromagnetic potential.
Now if we assume that magnetic monopoles exist, Maxwell's equations become \begin{align} \nabla\cdot\textbf{E}&=\frac{\rho_e}{\epsilon_0}\\ \nabla\cdot\textbf{B}&=\mu_0\rho_m\\ \nabla\times\textbf{E}&=-\mu_0\textbf{J}_m-\frac{\partial\textbf{B}}{\partial t}\\ \nabla\times\textbf{B}&=\mu_0\textbf{J}_e+\mu_0\epsilon_0\frac{\partial\textbf{E}}{\partial t} \end{align} The magnetic field is not divergenceless anymore. My question is how can we construct electromagnetic potential in that case.
