  We investigate boson star (BS) solutions in the Einstein-Proca theory with
the quartic order self-interaction of the vector field $\lambda
(A^\mu\bar{A}_\mu)^2/4$ and the mass term $\mu \bar{A}^\mu A_\mu/2$, where
$A_\mu$ is the complex vector field and ${\bar A}_\mu$ is the complex conjugate
of $A_\mu$, and $\lambda$ and $\mu$ are the coupling constant and the mass of
the vector field, respectively. The vector BSs are characterized by the two
conserved quantities, the Arnowitt-Deser-Misner (ADM) mass and the Noether
charge associated with the global $U(1)$ symmetry. We show that in comparison
with the case without the self-interaction $\lambda=0$, the maximal ADM mass
and Noether charge increase for $\lambda>0$ and decrease for $\lambda<0$. We
also show that there exists the critical central amplitude of the temporal
component of the vector field above which there is no vector BS solution, and
for $\lambda>0$ it can be expressed by the simple analytic expression. For a
sufficiently large positive coupling $\Lambda:=M_{pl}^2\lambda /(8\pi\mu^2) \gg
1$, the maximal ADM mass and Noether charge of the vector BSs are obtained from
the critical central amplitude and of ${\cal O}[\sqrt{\lambda}M_{pl}^3/\mu^2
\ln (\lambda M_{pl}^2/\mu^2)]$, which is different from that of the scalar BSs,
${\cal O}(\sqrt{\lambda_\phi}M_{pl}^3/\mu_\phi^2)$, where $\lambda_\phi$ and
$\mu_\phi$ are the coupling constant and the mass of the complex scalar field.
