Bank of America's recruitment committee is reported to have approached Mr Diamond but he is said to have spurned their advances to stay in his job with the UK bank.
The shortlist for the chief executive officer job at the US bank is believed to include Brian Moynihan, Bank of America's head of consumer and small-business banking, and Gregory Curl, chief risk officer at the bank.
Three external candidates are also believed to be on the recruitment committee's shortlist.
The job became available after current chief executive officer Kenneth Lewis announced last month that he is to take early retirement.
Controversy has dogged the final year of his reign, with much of it centred on the deal to buy Merrill Lynch in September 2008.
Earlier this month, Bank of America reported a quarterly loss for more than $1 billion.
By Gary Cooper