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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 $\sim5.4\times{10^4}$ progenitor sources, from which 21 were identified as candidate LRNe precursors. Further analysis revealed 16 of our candidates to be H$α$ 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 disks or alternatively magnetically active stellar merger remnants.