Purpose – The paper is aimed at the development of novel model reduction techniques for nonlinear systems. Design/methodology/approach – The analysis is based on the bilinear and polynomial representation of nonlinear systems and the exact solution of the bilinear system in terms of Volterra series. Two sets of Krylov subspaces are identified which capture the most essential part of the input-output behaviour of the system. Findings – The paper proposes two novel model-reduction strategies for nonlinear systems. The first involves the development, in a novel manner compared with previous approaches, of a reduced-order model from a bilinear representation of the system, while the second involves reducing a polynomial approximation using Krylov subspaces derived from a related bilinear representation. Both techniques are shown to be effective through the evidence of a standard test example. Research limitations/implications – The proposed methodology is applicable to so-called weakly nonlinear systems, where both the bilinear and polynomial representations are valid. Practical implications – The suggested methods lead to an improvement in the accuracy of nonlinear model reduction, which is of paramount importance for the efficient simulation of state-of-the-art dynamical systems arising in all aspects of engineering. Originality/value – The proposed novel approaches for model reduction are particularly beneficial for the design of controllers for nonlinear systems and for the design and analysis of radio-frequency integrated circuits.