The multimessenger detection of GW170817 showed that binary neutron star (BNS) mergers are progenitors of (at least some) short gamma-ray bursts (GRBs), and that short GRB jets (and their afterglows) can have structures (and observational properties) more complex than predicted by the standard top-hat jet scenario. Indeed, the emission from the structured jet launched in GW170817 peaked in the radio band (centimeter wavelengths) at ≈100 days since merger—a timescale much longer than the typical time span of radio follow-up observations of short GRBs. Moreover, radio searches for a potential late-time radio flare from the fast tail of the neutron-rich debris that powered the kilonova associated with GW170817 (AT 2017gfo) have extended to even longer timescales (years after the merger). In light of this, here we present the results of an observational campaign targeting a sample of seven, years-old GRBs in the Swift/BAT sample with no redshift measurements and no promptly identified X-ray counterpart. Our goal is to assess whether this sample of short GRBs could harbor nearby BNS mergers, searching for the late-time radio emission expected from their ejecta. We found one radio candidate counterpart for one of the GRBs in our sample, GRB 111126A, though an origin related to emission from star formation or from an active galactic nucleus in its host galaxy cannot be excluded without further observations.