We present a study of the core-collapse supernova (CCSN) rate in nuclei A and B1, of the luminous infrared galaxy (LIRG) Arp 299, based on similar to 11 yr of Very Large Array (VLA) monitoring of their radio emission at 8.4 GHz. Significant variations in the nuclear radio flux density can be used to identify the CCSN activity in the absence of high-resolution very long baseline interferometry (VLBI) observations. In the case of the B1-nucleus, the small variations in its measured diffuse (synchrotron plus free-free) radio emission are below the fluxes expected from radio supernovae (RSNe), thus making it well-suited to detect RSNe through flux density variability. In fact, we find strong evidence for at least three RSNe this way, which results in a lower limit for the CCSN rate (nu(SN)) of > 0.28(-0.15)(+0.27) yr(-1). This value agrees within the uncertainties with the infrared (IR) luminosity based SN rate estimate, and with previously reported radio estimates. In the A-nucleus, we did not detect any significant variability and found a SN detection threshold luminosity of approximate to 3.1 x 10(28) erg s(-1) Hz(-1), allowing only the detection of the most luminous RSNe known. Our method is basically blind to normal CCSN explosions occurring within the A-nucleus, which result in too small variations in the nuclear flux density, remaining diluted by the strong diffuse emission of the nucleus itself. Additionally, we have attempted to find near-IR (NIR) counterparts for the earlier reported RSNe in the Arp 299 nucleus A, by comparing NIR adaptive optics images from the Gemini-N Telescope with contemporaneous observations from the European VLBI Network (EVN). However, we were not able to detect NIR counterparts for the reported radio SNe within the innermost regions of nucleus A. While our NIR observations were sensitive to typical CCSNe at similar to 300 mas (or 70 pc projected distance) from the centre of the nucleus A, suffering from extinction up to A(V) similar to 15 mag, they were not sensitive to such highly obscured SNe within the innermost nuclear regions where most of the EVN sources were detected.