Context. The proper characterization of the least massive population of the young σ Orionis star cluster is required to understand the form of the cluster mass function and its impact on our comprehension of the substellar formation processes. S Ori 70 (T5.5 ± 1) and 73, two T-type cluster member candidates, are likely to have masses between 3 and 7 MJup if their age is 3 Myr. It awaits confirmation whether S Ori 73 has a methane atmosphere. Aims: We aim to: i) confirm the presence of methane absorption in S Ori 73 by performing methane imaging; ii) study S Ori 70 and 73 cluster membership via photometric colors and accurate proper motion analysis; and iii) perform a new search to identify additional T-type σ Orionis member candidates. Methods: We obtained HAWK-I (VLT) J, H, and CH4off photometry of an area of 119.15 arcmin2 in σ Orionis down to Jcomp = 21.7 and Hcomp = 21 mag. S Ori 70 and 73 are contained in the explored area. Near-infrared data were complemented with optical photometry using images acquired with OSIRIS (GTC) and VISTA as part of the VISTA Orion survey. Color-magnitude and color-color diagrams were constructed to characterize S Ori 70 and 73 photometrically, and to identify new objects with methane absorption and masses below 7 MJup. We derived proper motions by comparing of the new HAWK-I and VISTA images with published near-infrared data taken 3.4 - 7.9 yr ago. Results.S Ori 73 has a red H - CH4off color indicating methane absorption in the H-band and a spectral type of T4 ± 1. S Ori 70 displays a redder methane color than S Ori 73 in agreement with its latter spectral classification. Our proper motion measurements (μα cos δ = 26.7 ± 6.1, μδ = 21.3 ± 6.1 mas yr-1 for S Ori 70, and μα cos δ = 46.7 ± 4.9, μδ = -6.3 ± 4.7 mas yr-1 for S Ori 73) are larger than the motion of σ Orionis, rendering S Ori 70 and 73 cluster membership uncertain. From our survey, we identified one new photometric candidate with J = 21.69 ± 0.12 mag and methane color consistent with spectral type ≥ T8. Conclusions.S Ori 73 has colors similar to those of T3-T5 field dwarfs, which in addition to its high proper motion suggests that it is probably a field dwarf located at 170-200 pc. The origin of S Ori 70 remains unclear: it can be a field, foreground mid- to late-T free-floating dwarf with peculiar colors, or an orphan planet ejected through strong dynamical interactions from σ Orionis or from a nearby star-forming region in Orion.