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By combining Hubble Space Telescope (HST) and ground based optical and near-infrared photometric samples, we derive the RGB tip absolute magnitude of 22 galactic globular clusters (GGCs). The effects of varying the distance and the metallicity scales are also investigated. Then we compare the observed tip luminosities with those predicted by state-of-the-art stellar models that include the energy-loss due to the axion production in the degenerate core of red giant stars. We find that theoretical predictions including only the energy-loss by plasma neutrinos are, in general, in good agreement with the observed tip bolometric magnitudes, even though the latter are about 0.04 mag brighter, on the average. This small shift may be the result of systematic errors affecting the evaluation of the RGB tip bolometric magnitudes or, alternatively, it could be ascribed to an axion-electron coupling causing a non-negligible thermal production of axions. In order to estimate the strength of this possible axion sink, we perform a cumulative likelihood analysis using the RGB tips of the whole set of 22 GGCs. All the possible source of uncertainties affecting both the measured bolometric magnitudes and the corresponding theoretical predictions are carefully considered. As a result, we find that the value of the axion-electron coupling parameter that maximizes the likelihood probability is gae/10^13=0.60(+0.32;-0.58). This hint is valid, however, if the dominant energy sinks operating in the core of red giant stars are standard neutrinos and axions coupled with electrons. Any additional energy-loss process, not included in the stellar models, would reduce such a hint. Nevertheless, we find that values gae/10^13 > 1.48 can be excluded with a 95% of confidence.