  We examine the possibility of spontaneous vectorization in the vector-tensor
theories with the vector conformal and disformal couplings to matter. We study
the static and spherically symmetric solutions of the relativistic stars with
the nontrivial profile of the vector field satisfying the boundary conditions
$A_\mu\to 0$ at the spatial infinity, where $A_\mu$ represents the vector
field. First, we study the linear perturbations about the general relativistic
(GR) stellar solutions with the vanishing vector field $A_\mu =0$. We show that
the pure vector disformal coupling causes the ghost or gradient instability of
the GR stars, indicating the breakdown of the hyperbolicity on the GR stellar
backgrounds. On the other hand, the pure conformal coupling causes the
tachyonic instability of the GR solutions and would lead to the spontanenous
growth of the vector field toward the nontrivial solutions as in the manner of
spontaneous scalarization in the scalar-tensor theories. We then construct the
static and spherically symmetric solutions of the relativistic stars with $0$
and $1$ nodes of the vector field in the presence of the pure disformal
coupling. We find that the properties of the solutions with the nontrivial
vector field are similar to those of the generalized Proca theories. As in the
mass-radius diagram the branches of the $0$ and $1$ node solutions are
disconnected to that of the GR solutions in the lower density regimes, they may
be formed from the selected choice of the initial conditions. Finally, we
construct the $0$ node solutions in the case of the pure conformal coupling,
and show that the branch of these solutions is connected to that of the GR
solutions in both the low and high density regions and hence would arise
spontaneously via the continuous evolution from the GR solutions.
