Low frequency spin fluctuation dynamics in paramagnetic spinel LiV$_2$O$_4$, a rare 3$d$-electron heavy fermion system, is investigated. A parametrized self-consistent renormalization (SCR) theory of the dominant AFM spin fluctuations is developed and applied to describe temperature and pressure dependences of the low-$T$ nuclear spin-lattice relaxation rate $1/T_1$ in this material. The experimental data for $1/T_1$ available down to $sim 1$K are well reproduced by the SCR theory, showing the development of AFM spin fluctuations as the paramagnetic metal approaches a magnetic instability under the applied pressure. The low-$T$ upturn of $1/T_1T$ detected below 0.6 K under the highest applied pressure of 4.74 GPa is explained as the nuclear spin relaxation effect due to the spin freezing of magnetic defects unavoidably present in the measured sample of LiV$_2$O$_4$.
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