We study the time delay in the primary photoemission channel near the opening of an additional channel and compare it with the Wigner time delay in elastic scattering of the photoelectron near the corresponding inelastic threshold. The photoemission time delay near threshold is significantly enhanced, to a measurable 40 as, in comparison to the corresponding elastic scattering delay. The enhancement is due to the different lowest order of interelectron interaction coupling the primary and additional photoemission channels. We illustrate these findings by considering photodetachment from the H^{-} negative ion, and compare it with electron scattering on the hydrogen atom near the first excitation threshold. Other threshold processes of atomic photoionization and molecular photofragmentation, where photoemission time delay is enhanced, are identified. This opens the possibility of studying threshold behavior utilizing attosecond chronoscopy.