ZURICH—A group of Swiss banks is lining up to deliver key information to U.S. authorities investigating tax evasion, potentially moving a long-running legal battle further toward a close.

Last week, the Swiss cabinet unveiled a program that would allow roughly a dozen Swiss banks that are being investigated by the Justice Department to deliver data on Swiss bank accounts held by American clients that had been transferred to other institutions as U.S. authorities began cracking down on overseas tax evasion.

Under the program, banks would deliver "leaver lists"—aggregate data on bank accounts that can be used to identify tax evaders—directly to U.S. authorities, a step that could further chip away at Switzerland's historic banking privacy.

The program was developed after Switzerland's Parliament rejected a broader proposal for cooperating with U.S. authorities. It also allows banks to provide information on employees who assisted American clients in the past. Banks already being investigated by the DOJ have previously handed over such information to the U.S.

Some of the country's biggest banks, including Credit Suisse Group AG and Julius Baer Group AG , are eligible to apply or already have applied for Swiss government approval to participate, according to people familiar with the situation. Agreements between the banks and U.S. authorities are possible in the coming weeks, these people say.