We propose a new family of copulas generalizing the Farlie-Gumbel-Morgenstern (FGM) family and generated by two univariate functions. The main feature of this family is two permit the modeling of high positive dependence. In particular, it is established that the correlation range is [-3/4,1] and that the upper tail dependence coefficient can reach any value in [0,1]. Some examples of parametric subfamilies are provided.