We present the results of 75 As nuclear magnetic resonance (NMR), nuclear quadrupole resonance (NQR), and resistivity measurements in KFe2As2 under pressure (p). The temperature dependence of the NMR shift, nuclear spin-lattice relaxation time (T1) and resistivity show a crossover between a high-temperature incoherent, local-moment behavior and a low-temperature coherent behavior at a crossover temperature (T * ). T * is found to increase monotonically with pressure, consistent with increasing hybridization between localized 3d orbital-derived bands with the itinerant electron bands. No anomaly in T * is seen at the critical pressure pc = 1.8 GPa where a change of slope of the superconducting (SC) transition temperature Tc(p) has been observed. In contrast, Tc(p) seems to correlate with antiferromagnetic spin fluctuations in the normal state as measured by the NQR 1/T1 data, although such a correlation cannot be seen in the replacement effects of A in the AFe2As2 (A= K, Rb, Cs) family. In the superconducting state, two T1 components are observed at low temperatures, suggesting the existence of two distinct local electronic environments. The temperature dependence of the short T1s indicates nearly gapless state below Tc. On the other hand, the temperature dependence of the long component 1/T1L implies a large reduction in the density of states at the Fermi level due to the SC gap formation. These results suggest a real-space modulation of the local SC gap structure in KFe2As2 under pressure.Recent NMR measurements have pointed out a possible d-electron heavy fermion behavior in the AFe 2 As 2