We have used the IRAM 30-m and FCRAO 14-m telescopes to observe the molecular clumps associated with 12 ultracompact (UC) HII regions in the J=6-5, 8-7 and 13-12 rotational transitions of methyl-acetylene (CH3C2H). Under the assumption of LTE and optically thin emission, we have derived temperature estimates ranging from 30 to 56 K. We estimate that the clumps have diameters of 0.2-1.6 pc, H_2 densities of 10^5-10^6 {cm^{-3}}, and masses of 10^2-2 10^4 M_⊙. We compare these values with those obtained by other authors from different molecular tracers and find that the H_2 density and the temperature inside the clumps vary respectively like n_{H_2} ~ R^{-2.6} and T ~ R^{-0.5}, with R distance from the centre. We also find that the virial masses of the clumps are ~3 times less than those derived from the CH3C2H column densities: we show that a plausible explanation is that magnetic fields play an important role to stabilise the clumps, which are on the verge of gravitational collapse. Finally, we show that the CH3C2H line width increases for decreasing distance from the clump centre: this effect is consistent with infall in the inner regions of the clumps. We conclude that the clumps around UC HII regions are likely to be transient (~10^(5) yr) entities, remnants of isothermal spheres currently undergoing gravitational collapse: the high mass accretion rates (~10^{-2} M_⊙ yr^{-1}) lead to massive star formation at the centre of such clumps. Comment: 15 pages, 11 figures, A & A in press