The banks, which include Geneva-based Lombard Odier and Zurich firm EFG International (>> EFG International AG), came forward under a program brokered by the Swiss and U.S. governments, after criminal investigations of roughly a dozen Swiss banks including Credit Suisse (>> Credit Suisse Group AG) in the United States.
So-called second category banks will escape prosecution if they detail their wrong-doing with U.S. clients and pay fines under the program agreed last year.
But they must cooperate more fully with U.S. prosecutors before reaching non-prosecution agreements, Finanz und Wirtschaft reported on Saturday, citing unnamed legal sources.
Since a Dec. 31 deadline to enter the program, the banks have delivered information on how many American clients they have served in recent months, and clarified with U.S. officials what the penalties for harboring funds in offshore accounts will be.
In May, Zurich-based Credit Suisse became the largest bank in decades to plead guilty to a U.S. criminal charge and agreed to pay more than $2.5 billion in penalties for similar offences.
The U.S. Department of Justice, which last month extended the deadline for category 2 banks by one month, hasn't discussed specific financial penalties for those firms, according to Finanz und Wirtschaft.
Each bank's situation will be determined by September, the paper reported.
The banks are not required to hand over client names as part of the deal, but must commit to handing over information which will help U.S. officials set up judicial aid requests to pursue tax dodgers.
A spokesperson for the Swiss government wasn't immediately available for comment on Saturday.
(Reporting by Katharina Bart; Editing by Ruth Pitchford)