American Airlines and private equity firm TPG have made an offer to Japan Airlines which it says is "far superior" to offers made by other carriers.
The US carrier said that it has now offered Japan Airlines $1.1 billion (£659 million), which is more than the amount offered by rival airline Delta.
Aviation consultant at CRA International Mark Kiefer said: "There's a lot at stake here, especially given the importance of the Japanese market and the Asian market to all of these carriers."
He went on to say that the newest offer from American Airlines and TPG could see the fight for a stake in Japan Airlines drag on for some time.
Both American Airlines and Delta Airlines are enthusiastic about having a foot in the Asia-Pacific air travel market.
Japan Airlines has been struggling with massive debts for some time and has been bailed out four times by the government in the past eight years.
Written by Kimberley Homer.
Related news stories:Japan Airlines gets bid from consortium (18th September 2009)Japan Airlines announces US airline deal (13th January 2011)Japan Airlines 'to take American offer' (9th February 2010)New management take over Japan Airlines (3rd February 2010)