The purchase would give the company control of over 7,000 retail outlets around world selling vitamins, nutrition supplements and sports drinks.
Private equity firm Blackstone Group are also reported to be in talks to join the bid by for GNC. However, despite talks it is unclear whether Blackstone's participation in the deal is likely to happen.
News of Bright Food's bid for GNC first emerged earlier in the week, with sources saying talks were at an advanced stage and a deal could be announced within days.
The potential acquisition of Pittsburgh-based GNC, would follow a failed attempt by Bright Food to buy Britain's United Buscuits for about $3.2 billion.