Luminous red novae (LRNe) are astrophysical transients believed to be caused by the partial ejection of a binary star's common envelope (CE) and the merger of its components. The formation of the CE is likely to occur during unstable mass transfer, initiated by a primary star which is evolving off the main sequence (a Hertzsprung gap star) and a lower mass companion. In agreement with observations, theoretical studies have shown that outflows from the pre-CE phase produce a detectable brightening of the progenitor system a few years before the ejection event. Based on these assumptions, we present a method to identify Galactic LRNe precursors, the resulting precursor candidates, and our follow-up analysis to uncover their nature. We begin by constructing a sample of progenitor systems, i.e. Hertzsprung gap stars, by statistically modelling the density of a colour magnitude diagram formed from 'well behaved' Gaia DR2 sources. Their time-domain evolution from the Zwicky Transient Facility (ZTF) survey is used to search for slowly brightening events, as pre-CE precursor candidates. The nature of the resulting candidates is further investigated using archival data and our own spectroscopic follow-up. Overall, we constructed a sample of ~5.4 × 104 progenitor sources, from which 21 were identified as candidate LRNe precursors. Further analysis revealed 16 of our candidates to be Hα emitters, with their spectra often suggesting hotter (albeit moderately extincted) A-type or B-type stars. Because of their long-term variability in optical and mid-infrared wavelengths, we propose that many of our candidates are mass-transferring binaries with compact companions surrounded by dusty circumstellar discs or alternatively magnetically active stellar merger remnants.