Making the final decision on which agency wins will be Sir Colin Marshall, BA chairman, and Robert Ayling, group managing director.
Two other agencies, J Walter Thompson and Bartle Bogle Hegarty, have also been shortlisted, following all-day pitches to BA's marketing executives earlier this month.
It is estimated that the leading agencies have spent as much as £1m trying to win the business, underscoring the importance of the high-profile account.
New Saatchi is banking on the personal relationship between Maurice Saatchi and Sir Colin, who have been on good terms for years.
Maurice Saatchi, ousted from his former agency late last year, was a key figure behind BA's successful "World's Favourite Airline" campaign.
New Saatchi insiders say he is determined to win back the account.
Weighing against New Saatchi is the relatively small size of its current London-based team, 40 now, growing to 80 in August when New Saatchi takes on the Dixons account it won from Saatchi and Saatchi earlier this month - and the fact that its link to a global agency, Publicis, is so recent.
Bidding on the BA account requires worldwide support, an advantage that Saatchi & Saatchi, through holding company Cordiant, has in spades.
Another concern may be the running troubles between Publicis and True North, its American partner, which have at times bubbled over into acrimony.
Saatchi & Saatchi, which continues to handle the BA account pending the outcome of the review, has marshalled its whole BA team in the effort to retain the business.
If the account goes to New Saatchi, several more Saatchi & Saatchi executives are likely to jump ship.
A final decision from BA is expected early next month.Reuse content