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Static and dynamical magnetic and electronic properties of the kagome metal Sc$_3$Mn$_3$Al$_7$Si$_5$ have been investigated by $^{27}$Al nuclear magnetic resonance (NMR) measurements. The temperature dependence of Knight shift ($K$) shows a similar temperature dependence of the DC magnetic susceptibility $\chi$ except for the low-temperature region below $\sim$ 50 K where $K$ is almost constant while $\chi$ keeps increasing, which suggests that the increase in $\chi$ at low temperatures is not intrinsic. $^{27}$Al spin-lattice relaxation rate divided by temperature ($1/T_1T$) is found to be constant, confirming the metallic state of Sc$_3$Mn$_3$Al$_7$Si$_5$ from a microscopic point of view. Based on a Korringa ratio analysis using the $T_1$ and $K$ data, ferromagnetic spin fluctuations are found to dominate in Sc$_3$Mn$_3$Al$_7$Si$_5$. These fluctuations are suggested to be very slow with frequencies on the order of kilohertz or lower.