  Recently the LHCb collaboration reported a new exotic state $T^+_{cc}$ which
possesses $cc\bar u\bar d$ flavor structure. Since its mass is very close to
the threshold of $D^0D^{*+}$ (or $D^{*0}D^{+}$) and its width is very narrow,
it is inclined to conjecture that $T^+_{cc}$ is a molecular state of
$D^0D^{*+}$ (or $D^{*0}D^{+}$). In this paper we study the possible molecular
structures of $D^{(*)}D^{(*)}$ and $B^{(*)}B^{(*)}$ within the Bethe-Salpeter
(B-S) framework. We employ one boson exchange model to stand the interaction
kernels in the B-S equations. With reasonable input parameters we find the
isospin eigenstate $\frac{1}{\sqrt{2}}(D^0D^{*+}-D^{*0}D^{+})$ ($J^P=1^+$)
constitutes a solution, which supports the ansatz of $T^+_{cc}$ being a
molecular state of $D^0D^{*+}$ (or $D^{*0}D^{+}$). With the same parameters we
also find that the isospin-1 state
$\frac{1}{\sqrt{2}}(D^{*0}D^{*+}+D^{*0}D^{*+})$ ($J^P=0^+$) can exist.
Moreover, we also study the systems of $B^{(*)}B^{(*)}$ and their counterparts
exist as possible molecular states. Consistency of theoretical computations
based on such states with the data of the future experiments may consolidate
the molecular structure of the exotic state $T^+_{cc}$.
