Recently a new class of scalarized black holes in Einstein-Gauss-Bonnet (EGB) theories was discovered. What is special for these black hole solutions is that the scalarization is not due to the presence of matter, but it is induced by the curvature of spacetime itself. Moreover, more than one branch of scalarized solutions can bifurcate from the Schwarzschild branch, and these scalarized branches are characterized by the number of nodes of the scalar field. The next step is to consider the linear stability of these solutions, which is particularly important due to the fact that the Schwarzschild black holes lose stability at the first point of bifurcation. Therefore we here study in detail the radial perturbations of the scalarized EGB black holes. The results show that all branches with a nontrivial scalar field with one or more nodes are unstable. The stability of the solutions on the fundamental branch, whose scalar field has no radial nodes, depends on the particular choice of the coupling function between the scalar field and the Gauss-Bonnet invariant. We consider two particular cases based on the previous studies of the background solutions. If this coupling has the form used in [D. D. Doneva and S. S. Yazadjiev, Phys. Rev. Lett. 120, 131103 (2018)] the fundamental branch of solutions is stable, except for very small masses. In the case of a coupling function quadratic in the scalar field [H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou, and E. Berti, Phys. Rev. Lett. 120, 131104 (2018)], though, the whole fundamental branch is unstable.