In this paper, model reduction techniques for a class of nonlinear systems are proposed. Specifically, nonlinear systems are considered that can be decomposed as the feedback interconnection of a high-order linear subsystem and a nonlinear subsystem of relatively low order, allowing for the application of well-developed reduction techniques for linear systems. In this setting, conditions are given under which internal stability, as well as passivity or a bound on the L2 gain are preserved for the reduced-order nonlinear model. Additionally, a priori error bounds are given. In the derivation of the error bound, an incremental gain (or incremental passivity) property of the nonlinear subsystem is shown to be instrumental. Additionally, the techniques developed in this paper are applied in the scope of controller reduction, as is illustrated by means of an industrial temperature control benchmark example.