We investigate how the hyperfine coupling influences the NMR spin relaxation time, T1, in Weyl semimetals. Since the density of states in Weyl semimetals varies with the square of the energy around the Weyl point, a naive power counting predicts that the nuclear spin relaxation rate 1/T1 would depend on the temperature (T) and the chemical pontential (mu) as 1/T1  ∼ T max(T, mu)^4. We carefully investigate the hyperfine interaction between nuclear spins and Weyl fermions, and find that while its spin part behaves conventionally, its orbital part diverges unusually with the inverse of energy around the Weyl point. Consequently, in contrast with the naive estimate, we find 1/T1 ∼ T max(T, mu)^2 ln(max(T,mu)/ω0), where ω0 is the nuclear Larmor frequency. This allows us to identify an effective hyperfine coupling constant, which is tunable by gating or doping. We also analyze the recent experimental data on the nuclear spin-lattice relaxation rate of the Weyl semimetal TaP. We argue that its non-monotonic temperature dependence is explained by the temperature dependent chemical potential of Weyl fermions.