Motivated by the recent observation of ferromagnetic spin correlations in the filled skutterudite SrFe$_4$As$_{12}$ [Ding et al., Phys. Rev. B 98, 155149 (2018)], we have carried out $^{75}$As nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) measurements to investigate the role of magnetic fluctuations in a newly discovered isostructural superconductor SrOs$_4$As$_{12}$ with a superconducting transition temperature of $T_{rm c}$ $sim$ 4.8 K. Knight shift $K$ determined by the NQR spectrum under a small magnetic field ($le$ 0.5 T) is nearly independent of temperature, consistent with the temperature dependence of the magnetic susceptibility. The nuclear spin-lattice relaxation rate divided by temperature, 1/$T_1T$, is nearly independent of temperature above $sim$ 50 K and increases slightly with decreasing temperature below the temperature. The temperature dependence is reasonably explained by a simple model where a flat band structure with a small ledge near the Fermi energy is assumed. By comparing the present NMR data with those in SrFe$_4$As$_{12}$, we found that the values of $|K|$ and $1/T_1T$ in SrOs$_4$As$_{12}$ are smaller than those in SrFe$_4$As$_{12}$, indicating no obvious ferromagnetic spin correlations in SrOs$_4$As$_{12}$. From the temperature dependence of 1/$T_1$ in the superconducting state, an $s$-wave superconductivity is realized.
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