We present for the first time extended stellar density and/or surface brightness radial profiles for almost all the known Large Magellanic Cloud (LMC) old globular clusters (GCs). These were built from DECam images and reach out to ∼ 4 times the GCs' tidal radii. The background subtracted radial profiles reveal that the GCs located closer than ∼ 5 kpc from the LMC centre contain an excess of stars in their outermost regions with respect to the stellar density expected from a King profile. Such a residual amount of stars, not seen in GCs located farther than ∼ 5 kpc from the LMC centre, as well as the GCs' dimensions, shows a clear dependence with the GCs' positions in the galaxy, in the sense that, the farther the GC from the centre of the LMC, the larger both the excess of stars in its outskirts and size. Although the masses of GCs located inside and outside ∼ 5 kpc are commensurate, the outermost regions of GCs located closer than ∼ 5 kpc from the LMC centre appear to have dynamically evolved more quickly. These outcomes can be fully interpreted in the light of the known GC radial velocity disc-like kinematics, from which GCs have been somehow mostly experiencing the influence of the LMC gravitational field at their respective mean distances from the LMC centre.