This article proposes a high-gain observer design for a class of nonlinear systems with multiple known time-varying delays intervening in the states and the inputs. In the free delay case, the class of systems under consideration coincides with a canonical form characterising a class of multi-output nonlinear systems, which are observable for any input. The underlying high-gain design has been mainly motivated by its inherent simplicity from both design and implementation points of view. Indeed, the observer gain is determined from an explicit resolution of a time-invariant Lyapunov algebraic equation up to the specification of a single design parameter. An academic observation problem is addressed to illustrate the effectiveness of the proposed observer.