(Corrects description of a Fed advisory panel’s discussions in the seventh paragraph and updates Dudley’s status in a new 18th paragraph.)

A major financial crisis is brewing, brought on by a toxic combination of failing European sovereign governments and highly leveraged international banks, which could be affected by a breakup of the euro area.

In the weeks and months ahead, a major U.S. financial institution may face serious trouble, and government officials will have to decide whether that trouble is just a temporary lack of liquidity -- or a far more serious question of solvency.

In the original lender-of-last-resort formulation, more than 100 years ago, Walter Bagehot famously advised: “Lend freely at a high rate, on good collateral” when financial institutions hold valuable assets that cannot readily be converted into cash in the market. This doctrine has morphed in recent years, so that modern central banks think liquidity trouble requires them to provide essentially unlimited funding at a low rate, even if the collateral is dubious, as was done in 2008.

Central banks acted similarly last week by lowering the interest rate at which they will swap other currencies for dollars, the effect of which will make dollars available to European banks struggling with high levels of sovereign debt but without access to the Fed.

Insolvency, by contrast, means that a financial institution should be closed down because its assets are worth less than its debts (that is, shareholder capital isn’t enough to absorb the losses that the institution faces).

Liquidity versus insolvency is often a difficult judgment call, requiring rapid decision-making under great pressure and with limited information. Everyone likes a bailout -- and that’s what “lend freely at a low rate, on dubious collateral” has become. Smaller financial players aren’t likely to get the benefit of the doubt, but global megabanks are frequently pressing for special favors. The point of becoming “too big to fail,” after all, is to get protection from the central bank and other officials when downside risks materialize.

The Wall Street Journal reported last week that William C. Dudley, the president of the Federal Reserve Bank of New York, met in September with financial executives, who pressed for coordinated efforts by central banks to help alleviate Europe’s debt crisis. Last week, the Fed and five other central banks announced the availability of low-cost swap lines for financial companies, a form of coordinated action though not one the panel specifically suggested. There is nothing wrong with this kind of meeting, or with Fed officials like Dudley contacting financial players to learn more about market conditions.

Dudley, in fact, has done the public a service by creating an Investor Advisory Committee on Financial Markets, most of whose members do not work at “too big to fail” banks. Of the advisers’ institutions listed on the New York Fed’s website, I would regard only Credit Suisse Group as such a bank.

This brings at least some transparency to the Fed’s private-sector interactions. Still, many people think the Fed has traditionally tilted toward Wall Street banks. In the New York Fed’s 2007 annual report, for example, its directors included Jamie Dimon, the chairman and chief executive officer of JPMorgan Chase, and Richard Fuld, the chairman and CEO of Lehman Brothers Holdings Inc. Sandy Weill, the former chairman and CEO of Citigroup, was on the board from 2001 to 2006. Most board members, I should note, during the 2000s were not from “too big to fail” contenders.

The Fed has become increasingly sensitive to this issue, in part because of what happened with Bear Stearns, which JPMorgan ended up buying with some downside protection from the Fed. As for Lehman, the Fed and many other officials were slow to understand the dimensions of the financial system’s problems after the near-collapse of Bear Stearns. Then there’s Citigroup Inc., which under Weill’s successors received extraordinary support from the Fed and Treasury.

Perhaps the problem here is simply one of “optics” -- the governance of the New York Fed doesn’t look good from the outside. And the issue is not that Dudley has done anything wrong; to be clear, I am making no such accusation.

The legitimacy of top financial officials has increasingly come into question over the last three years. A recent article in Bloomberg Markets magazine quotes an unnamed fund manager alleging that Treasury Secretary Henry Paulson provided hedge funds with material, nonpublic information in July 2008, by describing in a private meeting how the U.S. could put Fannie Mae and Freddie Mac into conservatorship, which it did seven weeks later.

Bagehot’s “Lombard Street” was published in Britain in 1873, when the electoral franchise was still quite limited. Legitimacy of the lender-of-last-resort concept was not a major issue for him. But there is something inherently undemocratic about deciding behind closed doors who gets a generous bailout - - and who doesn’t.

This is why central banking is controversial in the U.S., and has been since the early 19th century. Presidents Thomas Jefferson and Andrew Jackson objected to the earliest form of central banking, which consisted of a powerful private bank with central-bank-type features.

The New York Fed has long had three kinds of directors: Class A represent banks that belong to the Federal Reserve system (and typically will be chief executive officers of such institutions); Classes B and C are supposed to be people who do not work at banks overseen by the Fed and represent agriculture, commerce, industry, services, labor and consumers. All three classes of directors once chose the New York Fed president, giving some the perception that the bank CEOs had too much say in the selection process.

The Dodd-Frank reform legislation recognized the sensitivity of this perceived conflict and mandated that only Class B and C directors could participate in choosing the regional Fed presidents.

The governance change was meant to strengthen the legitimacy of the New York Fed president. But Dudley, a former Goldman Sachs chief U.S. economist, was appointed by the board of the New York Fed under the rules in existence in January 2009, so the Class A directors presumably were very much involved.

Dudley was reappointed by Class B and C directors in December 2010; the New York Fed and the main Fed websites don’t mention this, nor does Dudley’s official biography. But my point remains: He was originally chosen by a board operating under pre-Dodd-Frank rules.

The Fed should attempt to address its perceived legitimacy problems, ahead of the impending crisis. Dudley should resign and make way for someone appointed under the new rules.

(Simon Johnson, who served as chief economist at the International Monetary Fund in 2007 and 2008, and is now a professor at the MIT Sloan School of Management and a senior fellow at the Peterson Institute for International Economics, is a Bloomberg View columnist. The opinions expressed are his own.)

To contact the author of this column: Simon Johnson at sjohnson@mit.edu.

To contact the editor responsible for this column: Paula Dwyer at pdwyer11@bloomberg.net.