A process and apparatus (1) for packaging a product (P) arranged on a tray (4) is disclosed. The apparatus comprises a packaging assembly (8) configured for tightly fixing one or more film sheets to one or more trays. The packaging assembly includes a lower tool (22) defining a prefixed number of seats (23) for receiving the trays and an upper tool (21) facing the lower tool and cooperating with the upper tool to define a packaging chamber (24). The packaging assembly (8) is configured to operate at least in a first operating condition, where said packaging chamber (24) is open, and in a second operating condition, where said packaging chamber (24) is hermetically closed. First injection apertures (90) are positioned in corner regions of the seat and are configured to inject streams of gas in a direction substantially parallel to and above the flange (4c) of a tray (4) positioned in said seat. The process includes the step of activating injection of a plurality of streams of gas through said first apertures (
