(Abridged) We aim to: i) confirm the presence of methane absorption in S Ori 73 (a T-type member candidate of the sig Orionis cluster, 3 Myr, 352 pc) through methane imaging; ii) study S Ori 70 and 73 cluster membership via photometric colors and accurate proper motion analysis; iii) perform a new search to identify additional T-type sig Orionis member candidates with likely masses below 7 Mjup. We obtained HAWK-I (VLT) J, H, and CH4off photometry of an area of 119.15 sq. arcmin in sig Orionis down to Jcomp = 21.7 and Hcomp = 21 mag. Near-infrared data were complemented with optical photometry using images acquired with OSIRIS (GTC) and VISTA as part of the VISTA Orion survey. We derived proper motions by comparison of the new HAWK-I and VISTA images with published near-infrared data taken 3.4 - 7.9 yr ago. 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 are larger than the motion of sig Orionis, rendering S Ori 70 and 73 cluster membership uncertain. We identified one new photometric candidate with J = 21.69 +/- 0.12 mag and methane color consistent with spectral type greater than T8. 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 sig Orionis or from a nearby star-forming region in Orion.
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