In this paper, a theory is proposed for a new optimal process to perform chained aggregation in reduced-order modelling. Notions of ‘almost invariant subspaces’ and ‘approximate aggregation’ are introduced in the proposed theory and are shown to be useful concepts in model reduction. To derive an exact aggregated model from the approximately aggregated model obtained by optimal chained aggregation, an algorithm is presented which is a variant of the well-known shifted QR algorithm. Examples are given to illustrate the theory and its superiority over a previous approach.