Earlier studies on hollow-circular rubber bearings, all of which are conducted for steel-reinforced bearings, indicate that the hole presence not only decreases the compression modulus of the bearing but also increases the maximum shear strain developing in the bearing due to compression, both of which are basic design parameters also for fiber-reinforced rubber bearings. This paper presents analytical solutions to the compression problem of hollow-circular fiber-reinforced rubber bearings. The problem is handled using the most-recent formulation of the "pressure method". The analytical solutions are,, then, used to investigate the effects of reinforcement flexibility and hole presence on bearing's compression modulus and maximum shear strain in the bearing in view of four key parameters: (i) reinforcement extensibility, (ii) hole size, (iii) bearing's shape factor and (iv) rubber compressibility. It is shown that the compression stiffness of a hollow-circular fiber-reinforced bearing may decrease considerably as reinforcement flexibility and/or hole size increases particularly if the shape factor of the bearing is high and rubber compressibility is not negligible. Numerical studies also show that the existence of even a very small hole can increase the maximum shear strain in the bearing significantly, which has to be considered in the design of such annular bearings.