The US government had hoped to arrange a bailout under which other US investment banks would finance a "bad bank" that would hold the most "toxic" investments of Lehman in the property and mortgage market.
The "good bank" or rest of the firm, including its investment and wealth management arms, would then be sold to another financial institution, for example Bank of America or the UK's Barclays.
Although such a deal would have cost the other investment banks millions, it might have restored confidence in the sector and avoided a sharp drop in the share price of all banks.
However, it appears that this plan is falling apart.
"The only thing that can prevent Lehman collapsing would be a huge injection of taxpayers' money," a banker close to the talks told the BBC, but added that US Treasury Secretary "Hank Paulson has made it clear he doesn't want to do that".
Bank of America, meanwhile, is said to be unconvinced that buying Lehman would be in the interest of its shareholders.
Instead, according to a report in the New York Times, Bank of America is in "advanced talks" to buy investment bank Merrill Lynch for more than $38bn.
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