The plan to impose the $60 annual fee has drawn sharp criticism, including comment from President Barack Obama, the Times reported Saturday.
BofA has said it would waive the fee for customers who held a mortgage with the bank or a $20,000 in their BofA accounts.
But the bank is considering new options for customers who use a BofA credit card, keep a minimum balance in their checking accounts or use direct deposits, the source said.
The nation's attitude toward banks as exemplified by the Occupy Wall Street movement is part of the reason BofA is looking over options to soften the blow to its customers, the Times said.
JPMorgan Chase & Co., imposed a $3 fee for using a debit card for customers in two states -- Georgia and Wisconsin -- to test the reaction. On Friday, the bank said it would not continue with the fee.
Citibank, as well, has said it would not charge a fee for debit cards, but said it would raise its fee for checking accounts.
Norma Garcia, head of the financial services program for Consumers Union, a division of Consumer Reports, said, "It's time for Bank of America to listen to its customers who are saying loud and clear, 'Drop the fee or we'll drop you."
"All banks that are considering debit card fees should ditch those plans," Garcia said.