The bid tops an offer from Australian life insurer AMP Ltd , which had faced resistance from AXA Asia Pacific's independent directors who were looking for a higher offer.
NAB's move would consolidate its position as Australia's top wealth manager, after its purchase of British insurer Aviva's Australian operations earlier this year.
NAB's bid came as AXA Asia Pacific weighed a sweetened takeover bid from Australia's AMP Ltd and French parent AXA SA , lobbed on Monday. That offer was worth $11.4 billion, or A$6.13 a share, based on AMP's close on Wednesday.
NAB said its proposal, conditional on support from AXA SA , would involve NAB first making an offer for all of AXA Asia Pacific, worth $11.98 billion in cash, and then on-selling the firm's Asian assets to AXA.
The proposal deliver Axa Asia-Pacific shareholders either A$6.43 cash per share or an alternative offer of A$1.59 cash and 0.1745 of a NAB share.
Under AMP's plan, AMP would also have bought the entire company and then on-sold the Asian assets to AXA SA.
Australia's biggest bank said its offer valued AXA's Australian and New Zealand arm at A$4.6 billion ($4.2 billion), compared with AMP's offer which as of Monday valued the same businesses at A$3.7 billion.
(Reporting by Jonathan Standing and Sonali Paul; Editing by Mark Bendeich)