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In an interview with CNBC earlier in the day, Trump said the central bank raised interest rates too fast, giving the Chinese the upper hand in trade negotiations.
"They devalue their currency; they have for years: It's put them at a tremendous competitive advantage. And we don't have that advantage because we have a Fed that doesn't lower interest rates," Trump told CNBC's Joe Kernen on "Squawk Box." "We should be entitled to have a fair playing field, but even without a fair playing field — because our Fed is very, very disruptive to us — even without a fair playing field we are winning."
"Don't forget: The head of the Fed in China is President Xi [Jinping] ... he can do whatever he wants," Trump said.
Summers, who served as Treasury secretary under President Bill Clinton and as an economic advisor to President Barack Obama, said the Fed should be "tilting towards easing" in the coming months. However, he takes issue with Trump's public pressure on the central bank, as well as what he sees as the president's admiration of dictators.
"I found the president's comments ... very disturbing. I think there is something depraved about his degree of dictator envy," he said on "Closing Bell. " "His idea that somehow the fact that we have an independent central bank ... is some kind of impediment, I think, is really a very, very dangerous idea."
Summers said Trump has a history of praising dictators or authoritarian leaders.
"What's tragic is that we've got a president who reaches out, admires, envies, celebrates and seeks to foster closer relationships with dictators while at the same time dissing, disregarding ... agreements and failing to build relationship with America's traditional allies," he said.
"Sometimes you can get your way in the short run with threats and extortion, but the ill will you leave behind ill serves you and ill serves the nation you lead."
Trump has broken with the tradition of past presidents, who generally refrained from weighing in on the direction of the Fed's monetary policy. He has repeatedly blasted Fed Chair Jerome Powell for hiking rates four times in 2018.
"It's more than just Jay Powell — we have people on the Fed that really weren't — they're not my people. But they certainly didn't listen to me because they made a big mistake: They raised interest rates far too fast," Trump told CNBC on Monday.
Summers calls the public pressure "counterproductive."
"The president's efforts to politicize the Fed will actually make it more difficult for the Fed to pursue easy policy because if they do there will inevitably be the sense that they're sacrificing their institutional independence to the wishes of the president."
The White House did not immediately respond to a request for comment on Summers' remarks.
— CNBC's Thomas Franck contributed to this report.