New measurements of the ^12^CO (1-0) and ^12^CO (3-2) line emission are presented for the nuclei of seven nearby starburst galaxies selected from a complete sample of 21 nearby starburst galaxies for which the nuclear star formation rates are measured to be comparable to the archetype starburst galaxies M82 and NGC 253. The new observations capitalize on the coincidence between the beam size of the 45 m Nobeyama telescope at 115 GHz and that of the 15 m James Clerk Maxwell Telescope at 345 GHz to measure the value of the ^12^CO (3-2)/(1-0) emission line ratio in a 15" (<2.5 kpc) diameter region centered on the nuclear starburst. In principle, the ^12^CO (3-2)/(1-0) emission line ratio provides a measure of temperature and optical depth for the ^12^CO gas. The error weighted mean value of the ^12^CO (3-2)/(1-0) emission line ratio measured for the seven starburst galaxy nuclei is 0.64+/-0.06. The ^12^CO (3-2)/(1 -0) emission line ratio measured for the starburst galaxy nuclei is significantly higher than the average value measured for molecular gas in the disk of the Galaxy, implying warmer temperatures for the molecular gas in starburst galaxy nuclei. On the other hand, the ^12^CO (3-2)/(1-0) emission line ratio measured for the starburst galaxy nuclei is not as high as would be expected if the molecular gas were hot, >20 K, and optically thin, τ < 1. The total mass of molecular gas contained within the central 1.2-2.8 kpc diameter region of the starburst galaxy nuclei ranges from 10^8^ to 10^9^ M_sun_. While substantial, the molecular gas mass represents only a small percentage, ~9%-16%, of the dynamical mass in the same region.