The bid trumps the 1.9 trillion yen offered by the Western Digital-led consortium, which also includes USA private equity firm KKR & Co LP. Western Digital's group is said to be working to increase its bid to 2 trillion yen.
Toshiba said that none of the bidders had taken pole position in the auction: "At this point, Toshiba has not made any decision to reduce the pool of candidate purchasers", the company said in a statement.
Alternatively, a combined Toshiba and Western Digital business could be nearly as large as Samsung's memory unit, giving it more negotiating leverage against Apple. Toshiba's main creditors, Innovation Network Corp. of Japan and the Development Bank of Japan, would then parter with South Korean chip manufacturer SK Hynix Inc to raise the 1.6 trillion Yen (£11.2 billion) needed to complete the purchase.
Selling the profitable chip division is seen as the key to cash-strapped Toshiba's survival as one of Japan's best-known firms struggles to plug multibillion-dollar losses at its US nuclear power division, Westinghouse Electric.
Toshiba is the world's second-biggest maker of memory chips after Samsung. Western Digital has also initiated legal action that threatens to derail any deal that does not have its consent. Yasuo Naruke, head of the chips business, is an outspoken critic of the Western Digital proposal, the people said, while the METI bureau involved in the talks is led by Tatsuya Terazawa.
Bain's revised bid was first reported by Japanese broadcaster NHK which said it would be structured so that Bain and Toshiba would each hold 46 percent of the unit.
The new offer comes as separate sources say the embattled Japanese conglomerate and Western Digital Corp are struggling to strike a deal ahead of their self-imposed deadline of Thursday. Representatives for Bain and Apple were not immediately available for comment. Toshiba and Western Digital combined could be as a large as Samsung's memory division.
Toshiba has been forced to sell its prized memory chip unit following heavy losses at its USA nuclear business unit Westinghouse that landed the company in a financial mess, involving billions of dollars in liabilities. "Highly reliable memory devices are essential to meet growing demand for storage".