This paper develops a functional reduced-order observer for a class of Linear Time-Invariant (LTI) systems with time-delay in the state. The proposed observer provides exponentially stable estimation errors. The unbiasedness conditions on the nominal part of the error dynamics of the reducedorder observer were given by employing an algebraic framework. Under such conditions, we propose the use of a partial pole placement of the error dynamical equations to handle sufficient conditions for the existence of the proposed observer. Specifically, we make use of the multiplicity-induced-dominancy property of the characteristic function corresponding to the system's error. The performance and the effectiveness of the developed observer is highlighted through two examples: first, the estimation of the HIV-1 infection dynamics, and second, the improvement of the environmental performance through the study of a diesel engine system.