The present paper examines the instability of a double-walled carbon nanotube embedded in an elastic medium under pure bending. Effect of surrounding elastic medium and van der Waals forces between the inner and outer nanotubes is taken into consideration. From the point of view of continuum modeling, an elastic double-shell model is presented for the pure bending buckling of a double-walled carbon nanotube. Based on this model, a condition is derived in terms of the buckling modes of the shell, from which the critical bending moment can be predicted. The paper emphasizes bifurcation instability. It is shown that buckling may occur. Finally, a simplified analysis is carried out to estimate the moment causing bifurcation instability of the double-walled carbon nanotube.