Context. Merging galaxy clusters allow for the study of different mass components, dark and baryonic, separately. Also, their occurrence enables to test the ΛCDM scenario, which can be used to put constraints on the self-interacting cross-section of the dark-matter particle.Aim. It is necessary to perform a homogeneous analysis of these systems. Hence, based on a recently presented sample of candidates for interacting galaxy clusters, we present the analysis of two of these cataloged systems.Methods. In this work, the first of a series devoted to characterizing galaxy clusters in merger processes, we perform a weak lensing analysis of clusters A1204 and A2029/A2033 to derive the total masses of each identified interacting structure together with a dynamical study based on a two-body model. We also describe the gas and the mass distributions in the field through a lensing and an X-ray analysis. This is the first of a series of works which will analyze these type of system in order to characterize them.Results. Neither merging cluster candidate shows evidence of having had a recent merger event. Nevertheless, there is dynamical evidence that these systems could be interacting or could interact in the future.Conclusions. It is necessary to include more constraints in order to improve the methodology of classifying merging galaxy clusters. Characterization of these clusters is important in order to properly understand the nature of these systems and their connection with dynamical studies.