Many Americans dread April 15, the deadline for filing their income tax returns. But some well-heeled people are trembling over another looming tax day: Oct. 15.

Thursday is the deadline for Americans to come clean about the money they have hidden offshore, in places like Swiss bank accounts. No one can say with certainty how much money is out there  the accounts are secret  but the hoard may be tens of billions of dollars.

Several thousand wealthy people have come forward, hoping to avoid large fines or possibly even prison. But many others are still weighing their options. The choice is stark: They can confess and pay the penalties, or gamble that they will not get caught. With the deadline only days away, tax lawyers say they are being inundated by anxious clients.

“We’re seeing a flood of people,” said Scott D. Michel, a tax lawyer in Washington. His firm, Caplin & Drysdale, has 350 clients who are preparing to report their offshore accounts to the Internal Revenue Service. The firm has 14 lawyers handling their cases, one of which involves a tax bill of hundreds of millions of dollars.