The anisotropy of the nuclear spin-lattice relaxation rate $1/T_{1}$ of $^{75}$As was investigated in the iron-based superconductor LaFeAs(O$_{1-x}$F$_{x}$) ($x = 0.07, 0.11$ and 0.14) as well as LaFeAsO. While the temperature dependence of the normal-state $1/T_1T$ in the superconducting (SC) $x = 0.07$ is different from that in the SC $x = 0.11$, their anisotropy of $1/T_1$, $R equiv (1/T_{1})_{H parallel ab}/(1/T_{1})_{H parallel c}$ in the normal state is almost the same ($simeq$ 1.5). The observed anisotropy is ascribable to the presence of the local stripe correlations with $Q = (pi, 0)$ or $(0, pi)$. In contrast, $1/T_1$ is isotropic and $R$ is approximately 1 in the overdoped $x = 0.14$ sample, where superconductivity is almost suppressed. These results suggest that the presence of the local stripe correlations originating from the nesting between hole and electron Fermi surfaces is linked to high-$T_c$ superconductivity in iron pnictides.
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