We thoroughly investigate conformally Schwarzschild spacetimes in different coordinate systems to seek for physically reasonable models of a cosmological black hole. We assume that a conformal factor depends only on the time coordinate and the spacetime is asymptotically flat Friedmann-Lemaître-Robertson-Walker universe filled by a perfect fluid obeying a linear equation state $p=w\rho$ with $w>-1/3$. In this class of spacetimes, the McClure-Dyer spacetime, constructed in terms of the isotropic coordinates, and the Thakurta spacetime, constructed in terms of the standard Schwarzschild coordinates, are identical and do not describe a cosmological black hole. In contrast, the Sultana-Dyer class of spacetimes, constructed in terms of the Kerr-Schild coordinates, describe a cosmological black hole and the corresponding matter field can be interpreted as a combination of a homogeneous perfect fluid and an inhomogeneous null fluid, which is valid everywhere in the spacetime unlike Sultana and Dyer's interpretation. The Culetu spacetime, constructed in terms of the Painlevé-Gullstrand coordinates, also describes a cosmological black hole and the corresponding matter field can be interpreted as a combination of a homogeneous perfect fluid and an inhomogeneous anisotropic fluid. It turns out, however, that in both the Sultana-Dyer and Culetu spacetimes, the total energy-momentum tensor violates all the standard energy conditions at a finite value of the radial coordinate in sufficiently late times.