We report a study of 139La(I=7/2) nuclear magnetic resonance (NMR) to investigate the structural phase transition in the two caged compounds LaIr2Zn20 and LaRu2Zn20 from microscopic viewpoints. The former compound shows a second-order structural phase transition around T s = 200 K whereas the latter one shows a first-order structural phase transition at T s = 150 K. For both compounds, quite narrow La-NMR lines of less that 10 kHz without any electric field gradient (EFG) at the La site were observed at temperatures above T s , indicating that the local symmetry at the La site holds a cubic symmetry. On the other hand, an EFG at La site appears at temperatures below T s , evidencing symmetry lowering occurs at the La site at temperatures below T s . At temperatures around T s , both the nuclear spin-lattice relaxation rate (1/T 1) and the nuclear spin-spin relaxation rate (1/T 2) show unusual enhancements associated with the slowing down of the EFG fluctuations. However, the temperature dependences of 1/T 1 T and 1/T 2 are different from the typical slowingdown phenomena of the EFG fluctuations due to the classical motional narrowing. The lowering of the local symmetry at the La site is ascribed to a freezing of the rattling motion.