Context. Supermassive stars (SMSs) with masses M?10(3)-10(4)M(?) formed by runaway collisions in young, massive, and dense star clusters have been invoked as a possible solution to the problem of the presence of multiple stellar populations and peculiar abundance patterns observed in globular clusters (GCs). However, no such objects have been observed so far.Aims. We aim to develop observational strategies to search for SMSs hosted within young massive clusters (thought to be the precursors of GCs) using both photometric and spectroscopic observations. Such strategies could be applicable in a relatively general fashion.Methods. We used theoretical predictions of the spectra of SMSs and SMS-hosting clusters, together with predictions from standard simple stellar populations to examine their impact on color-color diagrams and on individual optical spectral lines (primarily hydrogen emission and absorption lines). As a first step, we applied our search strategies to a sample of about 3000 young star clusters (YSCs) from two nearby galaxies with multiband observations from the HST and optical integral-field spectroscopy obtained with MUSE on the Very Large Telescope.Results. We focus on models for SMSs with large radii (corresponding to T-eff ? 7000 K), which predict strong Balmer breaks, and construct proper color-color diagrams to select the corresponding SMS-hosting cluster candidates. We show that the spectrophotometric properties of these latter are similar to those of normal clusters with ages of a few hundred million years. However, the cluster SEDs show signs of composite stellar populations due to the presence of nebular lines (Ha and others). Examining the photometry, overall SEDs, and the spectra of approximately 100 clusters with strong Balmer breaks, we find several objects with peculiar SEDs, the presence of emission lines, or other peculiar signatures. After careful inspection of the available data, we do not find good candidates of SMS-hosting clusters. In most cases, the composite spectra can be explained by multiple clusters or H II regions inside the aperture covered by the spectra, by contamination from a planetary nebula or diffuse gas, or by improper background subtraction. Furthermore, most of our candidate clusters are too faint to host SMSs.Conclusions. We demonstrate a strategy to search for SMSs by applying it to a sample of YSCs in two nearby galaxies. Our method can be applied to larger samples and also extended to higher redshifts with existing and upcoming telescopes, and therefore should provide an important test for GC-formation scenarios invoking such extreme stars.