We examine the physical conditions of the global molecular gas reservoir in the host galaxies of Seyfert 1 and Seyfert 2 nuclei. To do so, we acquired sensitive (12)CO, (13)CO, J = 1-0, and J = 2-1 observations and collected available data from the literature for a sample of 27 Seyfert galaxies. We find that Seyfert galaxies have an average value for the (12)CO/(13)CO J = 1-0 and J = 2-1 line ratios of [R(10)] = 12 and [R(21)] = 13, respectively, with no discernible dependence on the Seyfert type. The r(12) = (2-1)/(1-0) line ratio for (12)CO does not reveal any significant difference between the two types, but Seyfert galaxies as a class seem to have systematically lower values of r(12) (similar to 0.5-0.7) than do average spirals and starbursts. Moreover, for all the galaxies examined, but especially for Seyfert and starburst galaxies, we find that r(12) is likely to be smaller as the area of the galaxy sampled by the telescope beam becomes larger. This may be the consequence of a global gas excitation gradient in galaxies where warm (T(kin) greater than or similar to 20 K) gas lies confined, preferably in their central regions (less than or similar to 1 kpc), while a colder (T(kin) less than or similar to 10 K), and possibly subthermally excited, gas phase dominates the more extended CO emission in the disk. For Seyfert and starburst galaxies, there are indications that their gas excitation gradients may be quite similar. Examination of the properties of the molecular gas by using the r(12), R(10), and R(21) line ratios reveals that, unless r(12) less than or similar to 0.6, a single warm gas phase can account for the observed values of these ratios toward the central regions of an average Seyfert and/or starburst galaxy.