  The radio source at the center of the cool core galaxy cluster RBS 797
(z=0.35) is known to exhibit a misalignment of its radio jets and lobes
observed at different VLA-scale, with the innermost kpc-scale jets being almost
orthogonal to the radio emission which extends for tens of kpc filling the
X-ray cavities. Gitti et al. suggested that this peculiar radio morphology may
indicate a recurrent activity of the central radio source, where the jet
orientation is changing between the different outbursts due to the effects of
supermassive binary black holes (SMBBHs). We aim at unveiling the nuclear radio
properties of the brightest cluster galaxy (BCG) in RBS 797 and at
investigating the presence of a SMBBH system in its center. We have performed
new high-resolution observations at 5 GHz with the European VLBI Network (EVN),
reaching an angular resolution of 9x5 mas^2 and a sensitivity of 36
microJy/beam. We report the EVN detection of two compact components in the BCG
of RBS 797, with a projected separation of ~77 pc. We can envisage two possible
scenarios: the two components are two different nuclei in a close binary
system, or they are the core and a knot of its jet. Both interpretations are
consistent with the presence of SMBBHs. Our re-analysis of VLA archival data
seems to favor the first scenario, as we detect two pairs of radio jets
misaligned by ~90 degrees on the same kpc scale emanating from the central
radio core. If the two outbursts are almost contemporaneous, this is clear
evidence of the presence of two active SMBHs, whose radio nuclei are unresolved
at VLA resolution. The nature of the double source detected by our EVN
observations in the BCG of RBS 797 can be established only by future sensitive,
multi-frequency VLBI observations. If confirmed, RBS 797 would be the first
SMBBH system observed at medium-high redshift at VLBI resolution. (abridged)
