WASHINGTON — President Trump’s nonstop attacks on the Federal Reserve have raised eyebrows and broken with recent presidential norms. But an Aug. 23 tweet in which Mr. Trump called the Fed chair, Jerome H. Powell, an “enemy” of America appears to have prompted some hand-wringing inside the central bank.

“My only question is, who is our bigger enemy, Jay Powell or Chairman Xi?” Mr. Trump wrote, just moments after Mr. Powell wrapped up a speech at the monetary policy world’s top annual conference in Jackson, Wyo.

Mr. Trump, who himself nominated Mr. Powell to lead the Fed, had long complained about the central bank’s 2018 rate increases and had griped that officials were too slow in reversing course. But suggesting America’s most important economic leader was an enemy sent shock-waves through the economics profession — and spurred an email chain at the Fed itself.

Minutes after the president posted his tweet, the Fed’s communications director, Michelle Smith, sent an email to Richard Clarida, the Fed vice chair, with a screenshot of the tweets, based on documents released through a Freedom of Information Act request.