J.P. Morgan Chase is expected to pay more than $1 billion in penalties to the Justice Department to end a criminal probe into whether it provided adequate warnings about Bernard L. Madoff.

The deal, which would also include a deferred-prosecution agreement with U.S. Attorney Preet Bharara, could be wrapped up by the end of year, said others close to the case. Prosecutors have been looking for whether the bank failed to alert regulators despite numerous red flags. A central component of the case is why the bank didn't provide...