Moreover, Dick Fuld is also said to be facing a fight to keep the bank out of the clutches of other firms, as it struggles to recover from its credit crunch-related losses.
Lehman is reportedly considering selling off 70 per cent of its investment management business in order to raise revenue and shore up its balance sheet.
In exchange for the units, the bank is understood to be working to retain the right to buy them back at a later date.
However, as a necessary sweetener, a 20-25 per cent stake in the bank as a whole is understood to be on offer for the investment management business buyer - likely to be a private equity firm.
News of the potential deal comes alongside latest analysts' predictions, which forecast a further $4 billion of crunch-related losses for Lehman in the third quarter.
This would up the pressure on Mr Fuld, both on his own job and on the need to sell off the investment businesses.
Lehman refused to comment on the matter.