Four external candidates are already believed to have turned down the permanent post, including Barclays' president Bob Diamond and Citigroup director Michael O'Neill.
With Mr Lewis due to leave his post at the end of the year, Bank of America's board of directors is now considering the possibility of making an interim appointment if they cannot find a suitable candidate for the top job within the next three days, reports Bloomberg.
Matters are complicated further for the bank by the fact that any appointment will have to meet with the approval of US pay czar Kenneth Feinberg, who will also have the final say on the salary level of the new CEO.
Last week, the Wall Street Journal reported that one option may be to extend the tenure of Mr Lewis, who is believed to be willing to postpone his retirement until a replacement is found.
By Gary Cooper