This paper deals with high-gain nonlinear observer design for a class of triangular systems with delayed output measurements. Based on a recent high-gain like observer design method, called HG/LMI observer, a larger bound of the time-delay is allowed compared to that obtained by using the standard high-gain methodology. Such a HG/LMI observer leads to a significantly lower tuning parameter, which reduces the values of the observer gains and increases the maximum bound of the delay allowed to ensure exponential convergence. Indeed, an explicit relation between the maximum bound of the delay and the observer tuning parameter is inferred by using a Lyapunov-Krasovskii functional jointly with the Halanay inequality. Such a relation shows clearly the superiority of the use of the HG/LMI observer design methodology.