Context. The existence of supermassive black hole binaries (SMBHBs) is predicted by various cosmological and evolutionary scenarios for active galactic nuclei. These objects are considered as contributors into the gravitational wave (GW) background, as well as emitters of discrete GW bursts. Yet, SMBHBs remain a rather elusive class of extragalactic objects. Aims. Previously we have identified the quasar J2102+6015 as a potential SMBHB system on the basis of absolute astrometric very long baseline interferometry (VLBI) monitoring. Here we present another case, the source J0204+1514, exhibiting a similar oscillating astrometric pattern. Our aim is to analyse the evolution of SMBHBs as generators of GW and provide a physical 'multi-messenger'link between astrometric manifestation in the radio domain and GW emission. Methods. We analysed the available archive VLBI astrometry data that resulted in the detection of astrometric oscillations in the source J0204+1514. We assume these oscillations to be manifestations of orbital motion in a binary system. We estimated the parameters of the suspected SMBHB in this source and applied basic theoretical models to project its evolution towards coalescence. We also developed a simplified 'toy'model of SMBHBs consistent with the discovered astrometric oscillations and made quantitative predictions of GW emission of such sources using the case of J0204+1514 as an example potentially applicable to other SMBHBs. Results. We provide observational evidence of astrometric oscillations in the source J0204+1514. As an ad hoc result, we also provide a re-assessed estimate of the redshift of J2102+6015, z = 1.42. A toy model of the object containing a SMBHB with parameters consistent with the observed astrometric oscillations of the source J0204+1514 as an example enables us to consider GW emission as the cause of the system's orbital evolution. Conclusions. We conclude that astrometric VLBI monitoring has an appreciable potential for future detections of SMBHBs that could become multi-messenger targets for both electromagnetic (in radio domain) and GW astronomy. To outline the contours of a future physical model connecting SMBHB evolution with detectable GW manifestations, we present a toy model and, as an example, apply this toy model to the astrometrically oscillating source J0204+1514 described in this work. We also provide a justification for aiming future space-borne VLBI missions at direct imaging of SMBHBs as a synergistic contribution into future multi-messenger studies involving prospective GW facilities.