By using the Borel distribution series of the Mittag-Leffler type, we introduce a new class of the bi-close-to-convex functions defined in the open unit disk. We then apply the Faber polynomial expansion method in order to investigate the estimates for the general Taylor-Maclaurin coefficients of the functions belonging to this new class of bi-close-to-convex functions in the open unit disk. We consider the Fekete-Szegö type inequalities for the bi-close-to-convex function class and also consider several corollaries and the consequences of the results presented in this paper. © 2021 Journal of Nonlinear and Variational Analysis