Christine Lagarde, the president of the European Central Bank, at a recent conference described Jerome H. Powell, the chair of the Federal Reserve, as “the standard of the courageous central banker.”

The room was filled with central bank officials from around the world, who stood to applaud in a public outburst of support for a fellow central banker under relentless assault by President Trump.

Central banks’ independence, which allows policymakers to set interest rates free from political interference, is considered sacrosanct by investors and economists. That’s why the attacks on the Fed, the most important and influential central bank in the world, have reverberated far beyond Washington.

“It’s a signal which is worrisome,” said Stefan Ingves, the former governor of the Riksbank, Sweden’s central bank.

Turmoil in U.S. financial markets risks upending the finances of countries and companies abroad. The yields on long-term U.S. government debt, which serve as benchmarks for interest rates around the world, could spike if Mr. Trump were to fire Mr. Powell, economists warn. The value of the dollar could sink.

Many central banks hold a large share of their reserves in dollars and dollar-denominated assets. The dollar is also the dominant form of global payment, on one side of nearly 90 percent of all cross-border transactions, according to the Bank for International Settlements, the bank for central banks.