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[ABRIDGED] $\omega$ Centauri (NGC 5139) contains large numbers of variablestars of different types and, in particular, more than a hundred RR Lyraestars. We have conducted a variability survey of $\omega$ Cen in the NIR, usingESO's 4.1m Visible and Infrared Survey Telescope for Astronomy (VISTA). This isthe first paper of a series describing our results.  $\omega$ Cen was observed using VIRCAM mounted on VISTA. A total of 42 and100 epochs in $J$ and $K_{\rm S}$, respectively, were obtained, distributedover a total timespan of 352 days. PSF photometry was performed, and periods ofthe known variable stars were improved when necessary using an ANOVA analysis.  An unprecedented homogeneous and complete NIR catalogue of RR Lyrae stars inthe field of $\omega$ Cen was collected, allowing us to study, for the firsttime, all the RR Lyrae stars associated to the cluster, except 4 located faraway from the cluster center. Membership status, subclassifications betweenRRab and RRc subtypes, periods, amplitudes, and mean magnitudes were derivedfor all the stars in our sample. Additionally, 4 new RR Lyrae stars werediscovered, 2 of them with high probability of being cluster members. Thedistribution of $\omega$ Cen stars in the Bailey (period-amplitude) diagram isalso discussed. Reference lines in this plane, for both Oosterhoff type I (OoI)and II (OoII) components, are provided.  In the present paper, we clarify the status of many (candidate) RR Lyraestars that had been unclear in previous studies. This includes stars withanomalous positions in the color-magnitude diagram, uncertain periods or/andvariability types, and possible field interlopers. We conclude that $\omega$Cen hosts a total of 88 RRab and 101 RRc stars, for a grand total of 189 likelymembers. We confirm that most RRab stars in the cluster belong to an OoIIcomponent, as previously found using visual data.

An Updated Census of RR~Lyrae Stars in the Globular Cluster $\omega$~Centauri (NGC\,5139)