Aims: The detailed study of T-type candidate members of the young σ Orionis cluster (~3 Myr, ~352 pc, solar metallicity) is fundamental to properly assess the objects' cluster membership and their contribution to the definition of the substellar mass function. Methods: A total of three T-type candidates (S Ori 70, S Ori 73, and S Ori J053804.65-021352.5) lying in the line of sight towards σ Orionis were characterized by means of near-infrared photometric, astrometric, and spectroscopic studies. H-band methane images were collected for all three sources and an additional sample of 15 field T-type dwarfs using the LIRIS instrument on the 4.2 m William Herschel Telescope (WHT). J-band spectra of resolution of ~500 were obtained for S Ori J053804.65-021352.5 with the ISAAC spectrograph on the 8 m Very Large Telescope (VLT), and JH spectra of resolution of ~50 acquired with the Wide Field Camera 3 (WFC3) on board the Hubble Space Telescope (HST) were employed for the spectroscopic classification of S Ori 70 and 73. Accurate proper motions with a typical uncertainty of ±3 mas yr-1 and a time interval of ~7-9 yr were derived using old images and new data collected with ISAAC/VLT and WFC3/HST. Results: Using the LIRIS observations of the field T dwarfs, we calibrated this imager for T spectral typing via methane photometry. The three S Ori objects were spectroscopically classified as T4.5 ± 0.5 (S Ori 73), T5 ± 0.5 (S Ori J053804.65-021352.5), and T7 +0.5-1.0 (S Ori 70). These spectral types agree with the measured H-band methane colors. The similarity between the observed JH spectra and the methane colors and the data of field ultra-cool dwarfs of related classifications suggests that S Ori 70, 73, and S Ori J053804.65-021352.5 do not deviate significantly in surface gravity in relation to the field. Additionally, the detection of K I at ~1.25 μm in S Ori J053804.65-021352.5 points to a high-gravity atmosphere. Only the K-band reddish nature of S Ori 70 may be consistent with a low-gravity atmosphere. The proper motions of S Ori 70 and 73 are measurable and are larger than that of the cluster by >3.5σ. The proper motion of S Ori J053804.65-021352.5 is consistent with a null displacement. These observations suggest that none of the three T dwarfs is a likely σ Orionis member, and that either planetary-mass objects with masses below ~4 MJup may not exist free-floating in the cluster or they may lie at fainter near-infrared magnitudes than those of the targets (i.e., H > 20.6 mag), thus remaining unidentified to date. We determined the volume density of field T4-T7 dwarfs to be ≥2.8 ± 1.6 × 10-3 pc-3 from a survey that covered 2798.4 arcmin2 and was complete up to a distance of 119 pc.