The Farlie-Gumbel-Morgensten (FGM) family of bivariate distributions with given marginals, is frequently used in theory and applications and has been generalized in many ways. With the help of two auxiliary distributions, we propose another generalization and study its properties. After defining the rank of a distribution as the cardinal of the set of canonical correlations, we prove that some well-known distributions have practically rank two. Consequently we introduce several extended FGM families of rank two and study how to approximate any bivariate distribution to a simpler one belonging to this family.