An important open challenge in the area of computational argumentation is the automatic reconstruction of natural language enthymemes. Such argumentative figures are commonly used in natural language human discourse to improve the naturalness and efficiency of speech. They also represent a major challenge when developing computational argumentation systems that need to work with natural language data, since enthymemes bring irregularity to the representations proposed in classical models of argumentation. In this paper, we propose a new framework based on the theory of argumentation schemes aimed at automatically reconstructing natural language enthymemes. The proposed framework consists of a two-module pipeline: (i) scheme classification, and (ii) enthymeme reconstruction. We validate the proposed framework by comparing its performance to a baseline pipeline that does not take the argumentation scheme theory into account. We evaluate the framework by analysing the validity of the complete reconstructed arguments, establishing a new set of baselines that can be used as reference for future work in this direction.