Aims. We aim to use multiband imaging from the Phase-3 Verification Data of the J-PLUS survey to derive accurate photometric redshifts (photo-z) and look for potential new members in the surroundings of the nearby galaxy clusters A2589 (z = 0.0414) & A2593 (z = 0.0440), using redshift probability distribution functions (PDFs). The ultimate goal is to demonstrate the usefulness of a 12-band filter system in the study of largescale structure in the local Universe. Methods. We present an optimized pipeline for the estimation of photometric redshifts in clusters of galaxies. This pipeline includes a PSF-corrected photometry, specific photometric apertures capable of enhancing the integrated signal in the bluest filters, a careful recalibration of the photometric uncertainties and accurate upper-limit estimations for faint detections. To foresee the expected precision of our photo-z beyond the spectroscopic sample, we designed a set of simulations in which real cluster galaxies are modeled and reinjected inside the images at different signal-to-noise ratio (S/N) levels, recomputing their photometry and photo-z estimates. Results. We tested our photo-z pipeline with a sample of 296 spectroscopically confirmed cluster members with an averaged magnitude of ⟨r⟩ = 16.6 and redshift ⟨z⟩ = 0.041. The combination of seven narrow and five broadband filters with a typical photometric-depth of r ~ 21.5 provides δ_z/(1 + z) = 0.01 photo-z estimates. A precision of δ_z/(1 + z) = 0.005 is obtained for the 177 galaxies brighter than magnitude r < 17. Based on simulations, a δ_z/(1 + z) = 0.02 and δ_z/(1 + z) = 0.03 is expected at magnitudes ⟨r⟩ = 18 and ⟨r⟩ = 22, respectively. Complementarily, we used SDSS/DR12 data to derive photo-z estimates for the same galaxy sample. This exercise demonstrates that the wavelength-resolution of the J-PLUS data can double the precision achieved by SDSS data for galaxies with a high S/N. Based on the Bayesian membership analysis carried out in this work, we find as much as 170 new candidates across the entire field (~5 deg²). The spatial distribution of these galaxies may suggest an overlap between the systems with no evidence of a clear filamentary structure connecting the clusters. This result is supported by X-ray Rosat All-Sky Survey observations suggesting that a hypothetical filament may have low density contrast on diffuse warm gas. Conclusions. We prove that the addition of the seven narrow-band filters make the J-PLUS data deeper in terms of photo-z-depth than other surveys of a similar photometric-depth but using only five broadbands. These preliminary results show the potential of J-PLUS data to revisit membership of groups and clusters from nearby galaxies, important for the determination of luminosity and mass functions and environmental studies at the intermediate and low-mass regime.