We present Herschel Spectral and Photometric Imaging Receiver (SPIRE)-Fourier Transform Spectrometer (FTS) intermediate-sampled mapping observations of the central similar to 8 kpc (similar to 150 arcsec) of M51, with a spatial resolution of 40 arcsec. We detect four (CO)-C-12 transitions (J = 4-3 to J = 7-6) and the [C i] P-3(2)-P-3(1) and P-3(1)-P-3(0) transitions. We supplement these observations with ground-based observations of (CO)-C-12 J = 1-0 to J = 3-2 and perform a two-component non-local thermodynamic equilibrium analysis. We find that the molecular gas in the nucleus and centre regions has a cool component (T-kin similar to 10-20 K) with a moderate but poorly constrained density (n(H-2) similar to 10(3)-10(6) cm(-3)), as well as significant molecular gas in a warmer (T-kin similar to 300-3000 K), lower density (n(H-2) similar to 10(1.6)-10(2.5) cm(-3)) component. We compare our CO line ratios and calculated densities along with ratios of CO to total infrared luminosity to a grid of photon-dominated region (PDR) models and find that the cold molecular gas likely resides in PDRs with a field strength of G(0) similar to 10(2). The warm component likely requires an additional source of mechanical heating, from supernovae and stellar winds or possibly shocks produced in the strong spiral density wave. When compared to similar two-component models of other star-forming galaxies published as part of the Very Nearby Galaxies Survey (Arp 220, M82 and NGC 4038/39), M51 has the lowest density for the warm component, while having a warm gas mass fraction that is comparable to those of Arp 220 and M82, and significantly higher than that of NGC 4038/39.