Following the recent suggestion by Woosley & Hoffman (1992), we have pursued the possibility of r-process nucleosynthesis in the ''hot-bubble'' region of a type-II supernova, i.e. a region of high temperatures and low densities that is created by neutrino heating just outside the nascent neutron star. In contrast to previous works, we have used the results of detailed hydrodynamical simulations of the formation and evolution of the hot bubble as input for the nucleosynthesis calculations. The thermodynamical conditions in the hot-bubble region are discussed and their evolution with time is presented. Our nucleosynthesis calculations reveal that too many heavy nuclei with mass numbers A less than or similar to 90 are produced by the recombination of alpha-particles prior to the invoked r-processing. Namely, the lack of a sufficient number of free neutrons per seed nucleus prohibits the synthesis of the heavier elements by the neutron-capture process, which is in accordance with previous claims. The significant overproduction of elements with A almost-equal-to 90, in particular of Sr-88 , Y-89, and Zr-90, compared with observed solar system abundances, is interpreted as a clear indication that some essential physical aspect is still inadequately described in the model. By systematically exploring the variation of the alpha-processing with the neutron excess, with the density, and with the cooling timescale of the hot-bubble material, we find that a somewhat lower density in the hot-bubble region would enable a full r-process to occur. In a subsequent paper (Takahashi et al. 1994; Paper II), we demonstrate that an enchantingly neat r-processing is obtained by reducing the overall density in the hot-bubble region by a factor which is kept constant in space and time. We discuss the implications and meaning of such a density scaling in the context of the supernova models.