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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
preprint / 2009