We present a coherent and homogeneous multi-line study of the CS molecule in nearby (D$<$10Mpc) galaxies. We include, from the literature, all the available observations from the $J=1-0$ to the $J=7-6$ transitions towards NGC 253, NGC 1068, IC 342, Henize~2-10, M~82, the Antennae Galaxies and M~83. We have, for the first time, detected the CS(7-6) line in NGC 253, M~82 (both in the North-East and South-West molecular lobes), NGC 4038, M~83 and tentatively in NGC 1068, IC 342 and Henize~2-10. We use the CS molecule as a tracer of the densest gas component of the ISM in extragalactic star-forming regions, following previous theoretical and observational studies by Bayet et al. (2008a,b and 2009). In this first paper out of a series, we analyze the CS data sample under both Local Thermodynamical Equilibrium (LTE) and non-LTE (Large Velocity Gradient-LVG) approximations. We show that except for M~83 and Overlap (a shifted gas-rich position from the nucleus NGC 4039 in the Antennae Galaxies), the observations in NGC 253, IC 342, M~82-NE, M~82-SW and NGC 4038 are not well reproduced by a single set of gas component properties and that, at least, two gas components are required. For each gas component, we provide estimates of the corresponding kinetic temperature, total CS column density and gas density.
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