We present the results of a radio monitoring campaign to search for radio emission from nearby ultra-luminous X-ray sources (ULXs). These sources are bright off-nuclear X-ray point sources with luminosities exceeding LX > 1039 erg s-1. A well-defined sample of the 9 nearest ULXs has been monitored eight times over 5 months with the Very Large Array in A and B configuration. Our limiting sensitivity is ≈0.15 mJy (4σ) for radio flares and ≈60 μJy for continuous emission. In M 82 two ULXs seem to have coincident compact radio sources, which are probably supernova remnants. No continuous or flaring radio emission has been detected from any other ULX. Thus, ULXs do not generally emit steady-state radio emission above radio powers of 1.5 × 1017 W/Hz. The non-detections of the continuous emission are consistent with beamed or unbeamed radio emission from accreting black holes of ≤ 103 M⊙ based on the radio/X-ray correlation. Other published radio detections (M 82, NGC 5408) are also discussed in this context. Both detections are significantly above our detection limit. If ULXs have flaring radio emission above 4 × 1017 W/Hz we can give an upper limit on the duty cycle of the flares of 6%. This upper limit is in agreement with the observed number of flares in Galactic radio transients. Additionally we present a yet unreported radio double structure in the nearby low-luminosity AGN NGC 4736.