Context. OMC-2/3 is one of the nearest embedded cluster-forming regions that includes intermediate-mass protostars at early stages of evolution. A previous CO (3–2) mapping survey towards this region revealed outflow activity related to sources at different evolutionary phases. Aims. The present work presents a study of the warm gas in the high-velocity emission from several outflows found in CO (3–2) emission by previous observations, determines their physical conditions, and makes a comparison with previous results in low-mass star-forming regions. Methods. We used the CHAMP+ heterodyne array on the APEX telescope to map the CO (6–5) and CO (7–6) emission in the OMC-2 FIR 6 and OMC-3 MMS 1-6 regions, and to observe ¹³CO (6–5) at selected positions. We analyzed these data together with previous CO (3–2) observations. In addition, we mapped the SiO (5–4) emission in OMC-2 FIR 6. Results. The CO (6–5) emission was detected in most of the outflow lobes in the mapped regions, while the CO (7–6) was found mostly in the OMC-3 outflows. In the OMC-3 MMS 5 outflow, a previously undetected extremely high-velocity gas was found in CO (6–5). This extremely high-velocity emission arises from the regions close to the central object MMS 5. Radiative transfer models revealed that the high-velocity gas from MMS 5 outflow consists of gas with n_H2 = 10⁴–10⁵ cm⁻³ and T > 200 K, similar to what is observed in young Class 0 low-mass protostars. For the other outflows, values of n_H2 > 10⁴ cm⁻³ were found. Conclusions. The physical conditions and kinematic properties of the young intermediate-mass outflows presented here are similar to those found in outflows from Class 0 low-mass objects. Due to their excitation requirements, mid − J CO lines are good tracers of extremely high-velocity gas in young outflows likely related to jets.