The basic reproduction number $R_0$ of diseases depends on the population size (or the number of susceptible individuals). The commonly reported $R_0$ is based on what population size (when they do not mention about it)? Is there any standard population size for reporting $R_0$?
From what I understand (as an ecologist/population modeller), $R_0$ in epidemiology is not dependent on host population size per se, at least not in its basic form. It is also not dependent on the number of susceptible individuals, since it is defined as the number of secondary infections in a fully susceptible population, see e.g. this section in Farrington et al (2001):
The basic reproduction number of an infectious agent in a given population is the average number of secondary infections which one typical infected individual would generate if the population were completely susceptible.
This is also consistent with the Wikipedia description ("...in an otherwise uninfected population"). $R_0$ is however dependent on the environment (dispersal routes, host-host interactions etc), which also why it is used to evaluate and compare the effect of control measures. Host-host interactions is also dependent on population density, so $R_0$ is indirectly influenced by host population density, and this is partially why estimated $R_0$ values for the same disease will differ between countries and populations.
Fundamentally and conceptually, $R_0$ is the same thing as the Net reproductive rate (often also labelled $R_0$) in demography and population modelling, and it was originally borrowed from demography to epidemiology. As such, the net reproductive rate is often used in for instance pest control to evaluate the potential impact of pest species on agricultural crops, in direct analog to the basic reproduction number in epidemiology (see Emiljanowicz et al, 2014 for a randomly picked, recent example). Most commonly, $R_0$ (both in population dynamics and epidemiology) is calculated from static demographic rates, but nothing prevents you from considering stochastic effects, density dependence or e.g. ways that demographic rates are functions of host density or population size. This could lead to formulations where $R_0$ is an explicit function of host population density, but that is not the standard use of $R_0$.
To me, it seems like you are asking for the effective reproduction number ($R(t)$), which can be defined by:
...the average number of secondary cases that result from an infectious case in a particular population (Box 1). R depends on the level of susceptibility in the population, in contrast to the basic reproduction number (R0), which is the average number of secondary cases arising from one infectious case in a totally susceptible population."
This section is from a WHO paper on meases (Chiew et al, 2013). So basically, $R(t)$ is the average number of secondary cases per primary case at time $t$ (including the effects of immunity and/or control measures). As you can see, $R(t)$ is a function of time, and it can e.g. be used to describe the temporal dynamics of SIR-type models.