  We report on NMR studies of the quasi--1D antiferromagnetic $S=1/2$ chain
cuprate LiCuVO$_4$, focusing on the high--field spin--modulated phase observed
recently in applied magnetic fields $H > H_{\rm c2}$ ($\mu_0H_{\rm c2} \approx
7.5$ T). The NMR spectra of $^7$Li and $^{51}$V around the transition from the
ordered to the paramagnetic state were measured. It is shown that the
spin--modulated magnetic structure forms with ferromagnetic interactions
between spins of neighboring chains within the {\bf ab}--plane at low
temperatures 0.6 K $ < T < T_{\rm N}$. The best fit provides evidence that the
mutual orientation between spins of neighboring {\bf ab}--planes is random. For
elevated temperatures $T_{\rm N} < T \lesssim 15$ K, short--range magnetic
order occurs at least on the characteristic time scale of the NMR experiment.
