  In this paper we consider the following optimal stopping problem
$$V^{\omega}_{\rm A}(s) = \sup_{\tau\in\mathcal{T}}
\mathbb{E}_{s}[e^{-\int_0^\tau \omega(S_w) dw} g(S_\tau)],$$ where the process
$S_t$ is a jump-diffusion process, $\mathcal{T}$ is a family of stopping times
while $g$ and $\omega$ are fixed payoff function and discount function,
respectively. In a financial market context, if $g(s)=(K-s)^+$ or
$g(s)=(s-K)^+$ and $\mathbb{E}$ is the expectation taken with respect to a
martingale measure, $V^{\omega}_{\rm A}(s)$ describes the price of a perpetual
American option with a discount rate depending on the value of the asset
process $S_t$. If $\omega$ is a constant, the above problem produces the
standard case of pricing perpetual American options. In the first part of this
paper we find sufficient conditions for the convexity of the value function
$V^{\omega}_{\rm A}(s)$. This allows us to determine the stopping region as a
certain interval and hence we are able to identify the form of $V^{\omega}_{\rm
A}(s)$. We also prove a put-call symmetry for American options with
asset-dependent discounting. In the case when $S_t$ is a geometric L\'evy
process we give exact expressions using the so-called omega scale functions
introduced in Li and Palmowski (2018). We prove that the analysed value
function satisfies the HJB equation and we give sufficient conditions for the
smooth fit property as well. Finally, we present a few examples for which we
obtain the analytical form of the value function $V^{\omega}_{\rm A}(s)$.
