We derived potassium abundances in red-giant-branch stars in the Galactic globular clusters NGC 104 (144 stars), NGC 6752 (134 stars), and NGC 6809 (151 stars) using high-resolution spectra collected with FLAMES at the ESO – Very Large Telescope. In the samples we consider, we do not find significant intrinsic spreads in K/Fe, which confirms the previous findings, but which is at variance with the cases of the massive clusters NGC 2419 and NGC 2808. Additionally, marginally significant K/Fe−O/Fe anti-correlations are found in NGC 104 and NGC 6809, and K/Fe−Na/Fe correlations are found in NGC 104 and NGC 6752. No evidence of K/Fe−Mg/Fe anti-correlation are found. The results of our analysis are consistent with a scenario in which the process leading to the multi-populations in globular clusters also implies enrichment in the K abundance, the amplitude of the associated K/Fe enhancement becoming measurable only in stars showing the most extreme effects of O and Mg depletion. Stars enhanced in K/Fe have so far only been found in clusters harbouring some Mg-poor stars, while the other globulars, without a Mg-poor sub-population, show small or null K/Fe spreads.