Bipartite record linkage aims to link observations of the same individual across two distinct non-duplicated datasets. The two main approaches to solve this task are the Fellegi-Sunter model, which is based on comparing all pairs of observations; and the graphical record linkage model, which explicitly considers the data generating process and links observations to latent entities. In this work, we explore the similarities between these two methods. Specifically, we show that their parameters can be directly related under a common data model; and that they can be estimated within the same framework using a classification expectation-maximization algorithm, taking into account the problem constraints and allowing for the introduction of prior information.