BRUSSELS—International Monetary Fund chief Christine Lagarde is likely to become the first woman to run the European Central Bank, putting an experienced crisis fighter in charge and paving the way for a continuation of easy-money policies.

Ms. Lagarde also would be the institution’s first president without a pedigree in central banking. That has raised doubts about whether she would command the same credibility in financial markets as current chief Mario Draghi, who emerged as a dominant figure in the global economy during his nearly eight years at the ECB.

Her nomination comes as central bankers face challenges on a number of fronts. Inflation has weakened below target in many developed economies including the eurozone, while trade conflicts have crimped economic growth. But central bank rates are already super low or—in the case of Europe and Japan—negative, which spurs lending by reducing borrowing costs and making it unattractive to hold deposits..

Meanwhile, some central bankers are in the crosshairs of politicians. President Trump has routinely bashed Federal Reserve Chairman Jerome Powell for raising interest rates. And last month, he set his sights on Mr. Draghi, complaining that the ECB’s rate-cut signals were devaluing the euro. Days later, he praised the ECB president.

The ECB operates under tremendous scrutiny in Europe because the 19-member eurozone doesn’t have a common finance ministry, making the bank the region’s dominant voice on economic and monetary affairs.