We evaluate the ratio of the shear viscosity to the relaxation time of the shear current above but near the critical temperature T-c in SU(3) gauge theory on the lattice. The ratio is related to Kubo's canonical correlation of the energy-momentum tensor in Euclidean space with the relaxation-time approximation and an appropriate regularization. Using this relation, we evaluate the ratio on the lattice in Euclidean space. We obtain the ratio with reasonable statistics for the range of temperature 1.5 less than or similar to T/T-c less than or similar to 4. We find that the characteristic speed of the transverse plane wave propagating in a gluon medium v(eta) is almost constant, v(eta)(2) similar or equal to 0.5, for T/T-c greater than or similar to 1.5, which ensures the causality in this mode in second-order dissipative hydrodynamics.