  Cosmological models where dark matter (DM) and dark energy (DE) interact with
each other are the general scenarios in compared to the non-interacting models.
The interaction is usually motivated from the phenomenological ground and thus
there is no such rule to prefer a particular interaction between DM and DE.
Being motivated, in this work, allowing an exponential interaction between DM
and DE in a spatially flat homogeneous and isotropic universe, we explore the
dynamics of the universe through the constraints of the free parameters where
the strength of the interaction is characterized by the dimensionless coupling
parameter $\xi$ and the equation of state (EoS) for DE, $w_x$, is supposed to
be a constant. The interaction scenario is fitted using the latest available
observational data. Our analyses report that the observational data permit a
non-zero value of $\xi$ but it is very small and consistent with $\xi =0$. From
the constraints on $w_x$, we find that both phantom ($w_x< -1$) and
quintessence ($w_x> -1$) regimes are equally allowed but $w_x$ is very close to
`$-1$'. The overall results indicate that at the background level, the
interaction model cannot be distinguished from the base $\Lambda $-cold dark
matter model while from the perturbative analyses, the interaction model mildly
deviates from the base model. We highlight that, even if we allow DM and DE to
interact in an exponential manner, but according to the observational data, the
evidence for a non-zero coupling is very small.
