Alaska Airlines is understood to have made a bid valuing the company at more than $2bn.
The airline was believed to have been in competition with JetBlue Airways for the company.
The Sunday Times first reported that if the Alaska Airlines bid is accepted Branson, whose Virgin Group Holdings owns around 30 per cent of Virgin America's stock, would make nearly $700m.
Cyrus Capital Partners has a 28 per cent stake in Virgin America, meaning it could take a payout of around $600m.
Virgin America's share price has increased from less than $30 in mid-March to $38.90 on 1 April.
The potential sale comes after the airline sold stock in a $353m initial public offering, pricing its shares at $23 each, in 2014.
The California-based airline was launched in 2007.
In 2015 the company reported total operating revenue of $1.53bn, a 2.7 per cent rise on 2014.
Virgin, Alaska Airlines and JetBlue have been asked for comment.