* Western Digital CEO met Toshiba counterpart on Friday
* Company showed proposal to Toshiba recently -sources
* State-backed fund in talks with Bain Capital -sources (Adds details on Western Digital's proposal)
TOKYO, June 9 (Reuters) - Toshiba Corp said on Friday it was not convinced by Western Digital Corp CEO Steve Milligan's effort to win its backing for the U.S. tech firm's revised bid for the Japanese conglomerate's highly prized chip unit.
The failure of the two business partners to resolve tensions comes as uncertainty spikes over the make-up of the groups bidding for the world's second-largest producer of NAND memory chips, less than a week before Toshiba is to announce a winner.
Toshiba is rushing to find a buyer for the business, which it values at $18 billion or more, to cover billions of dollars in cost overruns at its now-bankrupt U.S. nuclear business Westinghouse Electric Corp.
Toshiba is aiming to announce its pick of a preferred bidder on Thursday.
Some sources see Western Digital, which jointly operates a key flash memory chip plant with Toshiba in Japan, as key to any successful deal.
The two firms are, however, at loggerheads over the auction, with Western Digital pursuing an international arbitration claim that says Toshiba has breached the joint venture contracts by entertaining outside bids, and arguing that the sale cannot proceed without its consent.
Milligan and Toshiba CEO Satoshi Tsunakawa met at Toshiba headquarters on Friday afternoon.
"Toshiba listened to Western Digital's thinking, but our concerns about the prospects of success for a deal were not wiped out," a Toshiba spokeswoman said.
A Western Digital spokeswoman confirmed that the meeting took place but declined to comment further.
Western Digital on Wednesday proposed a consortium led by a Japanese state-backed fund, the Innovation Network Corp of Japan, and including U.S. private equity firm KKR & Co, sources told Reuters, declining to be identified as they were not authorized to talk to the media.
A source familiar with Western Digital said the revised proposal included a significantly increased deal value as well as an investment in Toshiba Memory's convertible debt that could help smooth the regulatory process.
The Japan-backed consortium also has commitments in place that Toshiba's essential technology will stay in Japan, the source added.
This group is competing with U.S. chipmaker Broadcom Ltd , which has partnered with U.S. private equity firm Silver Lake.
Toshiba favors the Broadcom bid as it is higher and would avoid antitrust concerns triggered by Western Digital, which overlaps heavily with Toshiba on NAND chips.
A senior Toshiba executive said after Friday's meeting that even a minority stake by Western Digital would make it hard to clear antitrust review.
In its bid, Western Digital initially sought a majority but has agreed to limit its stake to 19.9 percent to appease Japan's government. The bid price is still short of Toshiba's threshold 2 trillion yen ($18 billion) and lower than Broadcom's offer of 2.2 trillion yen, they said.
Attention is now focused on last-ditch efforts to salvage the government-Western Digital bid.
Two sources familiar with the matter said on Friday the Japanese state-backed fund was in talks with U.S. private equity firm Bain Capital about teaming up in a bid. One of the sources said Bain would replace KKR as the main partner for the state-backed fund. In the previous round of bidding, Bain partnered with South Korean chipmaker SK Hynix. ($1 = 110.6800 yen) (Reporting by Makiko Yamazaki and Junko Fujita; Additional reporting by Yoshiyasu Shida and Taro Fuse in Tokyo and Se Young Lee in Seoul and Liana B. Baker in San Francisco; Editing by Edwina Gibbs and Paul Simao)