Nippon Life was likely to pay 100 billion to 200 billion yen ($887 million to $1.77 billion) for the stake.
Bain’s consortium also includes South Korean chipmaker SK Hynix, as well as Dell Inc, Seagate Technology Plc and Kingston Technology.
It's unclear if the sale will proceed smoothly as Western Digital has previously initiated legal action against Toshiba.
Toshiba aims to sign a contract by late September but added that the memorandum does not exclude talks with other bidders.
IPOs & Markets
Toshiba's board, under pressure to clinch a deal soon, met to consider Western Digital's new proposal.
The move may help Toshiba finally seal a deal to sell its prized unit after months of delays.
Toshiba is continuing to talk with all three suitors and has not made any decision to reduce the pool of candidates.
Apple, a key client for Toshiba, is a new member of the consortium, which also includes South Korean chipmaker SK Hynix Inc.
Other members of group, which is offering $17.4b, will include INCJ, the Development Bank of Japan as well as KKR.
Toshiba is scrambling to sell its flash memory unit to cover losses from its bankrupt US nuclear business Westinghouse.
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