Core collapse supernovae (CCSNe) produce fast shocks which pervade the dense circum-stellar medium (CSM) of the stellar progenitor. Cosmic rays (CRs) if accelerated at these shocks can induce the growth of electromagnetic fluctuations in the foreshock medium. In this study, using a self-similar description of the shock evolution, we calculate the growth timescales of CR-driven instabilities. We select a sample of nearby core collapse radio supernova of type II and Ib/Ic. From radio data we infer the parameters which enter in the calculation of the instability growth times. We find that extended IIb SNe shocks can trigger fast intra day instabilities, strong magnetic field amplification and CR acceleration. In particular, the non-resonant streaming instability can contribute to about 50% of the magnetic field intensity deduced from radio data. This results in the acceleration of CRs in the range 1-10 PeV within a few days after the shock breakout. In order to produce strong magnetic field amplification and CR acceleration a fast shocks pervading a dense CSM is necessary. In that aspect IIn supernovae~are also good candidates. But a detailed modeling of the blast wave dynamics coupled with particle acceleration is mandatory for this class of object before providing any firm conclusions. Finally, we find that the trans-relativistic object SN 2009bb even if it produces more modest magnetic field amplification can accelerate CRs up to 2-3 PeV within 20 days after the outburst.
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