In this paper, we consider the problem of designing reduced-order linear functional interval observers for nonlinear uncertain time-delay systems with external unknown disturbances. Given bounds on the uncertainties, we design two reduced-order linear functional state observers in order to compute two estimates, an upper one and a lower one, which bound the unmeasured linear functions of state variables. Conditions for the existence of a pair of reduced-order linear functional observers are presented, and they are translated into a linear programming (LP) problem in which the observers' matrices can be effectively computed. Finally, the effectiveness of the proposed design method is supported by four examples and simulation results.