The Money Laundering Deterrence Act was approved by the House Monday.
The legislation would let banks share information about former employees suspected of illegal activity and still be shielded from civil liability as long as the statements were not reckless or driven by malice.
It is not clear whether the bill is a high enough priority for the Senate to act on it before adjourning Friday.
But John J. Byrne, senior counsel at the American Bankers Association, which supports the legislation, said the Senate is "inclined to move it." A spokesman for the House Banking Committee agreed.
The anti-money-laundering bill would also require federal bank, thrift, credit union, and nonbank supervisors to issue "know-your-customer" rules for comment by early February.