$^{139}$La nuclear magnetic resonance (NMR) measurements under pressure ($p = 0-2.64$ GPa) have been carried out to investigate the static and dynamic magnetic properties of the itinerant ferromagnet LaCrGe$_3$. $^{139}$La-NMR spectra for all measured pressures in the ferromagnetically ordered state show a large shift due to the internal field induction $|$$B_{rm int}$$|$ $sim$ 4 T at the La site produced by Cr ordered moments. The change in $B_{rm int}$ by less than 5% with $p$ up to 2.64~GPa indicates that the Cr 3$d$ moments are robust under pressure. The temperature dependence of NMR shift and $B_{rm int}$ suggest that the ferromagnetic order develops below $sim$ 50~K under higher pressures in a magnetic field of $sim$ 7.2 T. Based on the analysis of NMR data using the self-consistent-renormalization (SCR) theory, the spin fluctuations in the paramagnetic state well above $T_{rm C}$ are revealed to be three dimensional ferromagnetic throughout the measured $p$ region.
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