  In this paper, we study a Dirichlet type problem for the non-pluripolar
complex Monge - Amp\`ere equation with prescribed singularity on a bounded
domain of $\mathbb{C}^n$. We provide a local version for an existence and
uniqueness theorem proved by Darvas, Di Nezza and Lu. Our work also extends a
result of Ahag, Cegrell, Czyz and Pham.
