  In this paper we investigate the case of non-minimal coupling in the
(extended) nonlinear massive gravity theories. We first consider massive
gravity in the Brans-Dicke background such that the graviton mass is replaced
by $A^2(\sigma)m$ where $\sigma$ is the Brans-Dicke field and $A(\sigma)$ is
conformal coupling and show that there is no viable thermal history of the
universe in this case. We then invoke a cubic galileon term as nonlinear
completion of the $\sigma$ Lagrangian and show that there is a stable de Sitter
solution in this case. However, the de Sitter is blocked by the matter phase
which is also a simultaneous attractor of the dynamics. The de Sitter phase
can, however, be realized by invoking unnatural fine tunings. We next
investigate cosmology of quasi-dilaton nonlinear massive gravity with
non-minimal coupling. As a generic feature of the non-minimal coupling, we show
that the model exhibits a transient phantom phase which is otherwise
impossible. While performing the observational data analysis on the models, we
find that a small value of coupling constant is allowed for quasi-dilaton
nonlinear massive gravity. For both the cases under consideration, it is
observed that we have an effective pressure of matter which comes from the
constraint equation. For mass-varying nonlinear massive gravity in the
Brans-Dicke background, the effective pressure of matter is non zero which
affects the evolution of the Hubble parameter thereby spoiling consistency of
the model with data. As for, quasi-dilaton nonlinear massive gravity, the
effective pressure of matter can be kept around zero by controlling the
coupling constant, the model is shown to be fit well with observations.
