"Don't be a fool," Donald Trump warned his counterpart Recep Tayyip Erdogan in an extraordinary letter sent the day Turkey launched its incursion into northeastern Syria — warning history risked branding him a "devil".

Read: US to withdraw troops as Turkey launches Syria operation: White House

Three days after appearing to greenlight an invasion by pulling US troops from the Kurdish-dominated region, Trump told the Turkish president he would wreck Ankara's economy if the invasion went too far.

In language shorn of diplomatic niceties, Trump began with an outright threat.

"Let's work out a good deal," Trump wrote in the letter dated October 9.

"You don't want to be responsible for slaughtering thousands of people, and I don't want to be responsible for destroying the Turkish economy — and I will."

"History will look upon you favorably if you get this done the right and humane way," Trump said.

"It will look upon you forever as the devil if good things don't happen."

The US leader told Erdogan a "great deal" was possible if he negotiated with the head of the Kurdish-led Syrian Democratic Forces, Mazloum Abdi, whom Turkey has labelled a "terrorist" for his ties to the Kurdish PKK militants in Turkey.

"Don't be a tough guy. Don't be a fool," he finished, adding: "I will call you later."