Banks typically welcome deposits as cheap funding that they redeploy into loans and investments, but BNY said the recent flood of cash is affecting its capital ratio and insurance fees.

"The transient nature of these new deposits prevents us from investing our balance sheet to cover the costs incurred from sudden and significant increases in U.S. Dollar Deposits with BNY Mellon," the bank said in a letter sent this week to clients.

The bank urged clients to reduce their deposit balance and "to consider a variety of cash investment options to minimize any effect on you."

The fee will apply to certain accounts holding greater than $50 million "per client relationship."

Traders cited this as fueling demand for U.S. Treasury billsand pushing one-month bill rates near zero.

BNY Mellon said the fee of 0.13 percentage point will take effect on Aug. 8 on certain deposits whose monthly average balance is greater than $50 million.

"Recent market events such as the Greek debt crisisand the uncertainty created by the handling of the U.S. debt ceiling have caused many of our clients to alter their cash management strategies," BNY Mellon said.