It is well-known that observer design is a powerful tool to estimate the states of a dynamical system. Given a multi-output nonlinear dynamical system whose states are partially observable, this paper investigates the problem of observer design to estimate those observable states. It considers firstly a nonlinear system without inputs, and provides a set of geometric conditions, guaranteeing the existence of a change of coordinates which splits the studied nonlinear dynamical system into two subsystems, where one of them is of the well-known nonlinear observer normal form, for which a Luenberger-like observer can be designed. An extension to nonlinear systems with inputs has then been deduced.