The Large Area Telescope (LAT) on board the Fermi Gamma-ray Space Telescope (Fermi) shows long-lasting high-energy emission in many gamma-ray bursts (GRBs), similar to X-ray afterglows observed by the Neil Gehrels Swift Observatory (Swift). Some LAT light curves (LCs) show a late-time flattening reminiscent of X-ray plateaus. We explore the presence of plateaus in LAT temporally extended emission analyzing GRBs from the second Fermi-LAT GRB Catalog from 2008 to 2016 May with known redshifts, and check whether they follow closure relations corresponding to four distinct astrophysical environments predicted by the external forward shock model. We find that three LCs can be fit by the same phenomenological model used to fit X-ray plateaus and show tentative evidence for the existence of plateaus in their high-energy extended emission. The most favorable scenario is a slow-cooling regime, whereas the preferred density profile for each GRBs varies from a constant-density interstellar medium to an r⁻² wind environment. We also compare the end time of the plateaus in γ-rays and X-rays using a statistical comparison with 222 Swift GRBs with plateaus and known redshifts from 2005 January to 2019 August. Within this comparison, the case of GRB 090510 shows an indication of chromaticity at the end time of the plateau. Finally, we update the 3D fundamental plane relation among the rest-frame end time of the plateau, its correspondent luminosity, and the peak prompt luminosity for 222 GRBs observed by Swift. We find that these three LAT GRBs follow this relation.