European banks have rushed to cut deals with prosecutors over longstanding claims that they pushed toxic mortgage securities in the years before the financial crisis.

The payouts are steep: Deutsche Bank and Credit Suisse said that they would disgorge nearly $13 billion combined to settle with the United States Justice Department.

But with the clock ticking before President-elect Donald J. Trump takes over, there appears to be an eagerness in Washington to conclude cases before a new, potentially more sympathetic, administration begins. As a result, these banks may have benefited from paying billions less than once proposed.

The $7.2 billion settlement with Deutsche Bank was a relief on Friday to its investors, who were rattled when it emerged in September that prosecutors were seeking a penalty of as much as $14 billion. Shares of Deutsche Bank rose as much as 5 percent in Frankfurt, before settling up 0.8 percent.