In this paper, transversal flow field of nonlinear viscoelastic fluids abiding by the modified-Phan-Thien-Tanner (MPTT) constitutive model in straight tubes of eccentric-annular cross-section is investigated. An analytical solution is developed based on an asymptotic expansion in terms of the Weissenberg number coupled with the shape factor method a one-to-one mapping taking the circular cross-section into the eccentric annular cross section. The analysis reveals the formation of transversal flows due to elasticity and to the eccentricity parameter. The number of vortices in the cross-section depends on the ratio of the diameters in addition to the eccentricity parameter. The effect of these parameters on the vortical structure is explored for different values of the material parameters.