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The October 9 letter was released by the White House on Wednesday as Trump battled to control the political damage following his decision to pull US troops out of northern Syria, clearing the way for the Turkish incursion against America's Kurdish allies.
The letter tried to persuade Erdogan to reverse a decision to invade Syria that Erdogan told Trump about in an October 6 phone call.
"Let's work out a good deal!" Trump said. "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."
Trump had the letter released to bolster his view that he did not give Turkey a green light to invade Syria. Many lawmakers have been sharply critical of his decision to remove American forces from the conflict zone.
"I have worked hard to solve some of your problems. Don't let the world down. You can make a great deal," said Trump in the letter.
The president wrote that the commander of the Kurdish-led Syrian Democratic Forces, General Mazloum Kobani Abdi, was willing to negotiate and to make some concessions.
He said he had confidentially enclosed to Erdogan a copy of a letter Mazloum had sent him.
"History will look upon you favourably if you get this done the right and humane way. It will look upon you forever as the devil if good things don't happen. Don't be a tough guy. Don't be a fool!" said Trump.
He added: "I will call you later."
Meanwhile, Trump said he had no problem if Russia helped Syria in a conflict with NATO ally Turkey, rejecting criticism of his withdrawal of US troops from Syria that exposed Kurdish allies, calling it "strategically brilliant."
Washington's hasty exit has created a land rush between Turkey and Russia - now the undisputed foreign powers in the area - to partition the formerly US-protected Kurdish area.
Syrian troops accompanied by Russian forces entered the city of Kobani, a strategically important border city and a potential flashpoint for a wider conflict, the UK-based Syrian Observatory for Human Rights, which monitors the Syrian war, reported.
Speaking to reporters on Wednesday, Trump said the Kurds were "not angels" and that it might be necessary for Russian-backed Syria and Turkey to "fight it out."
In the Oval Office he said: “Our soldiers are not in harm’s way - as they shouldn’t be, as two countries fight over land that has nothing to do with us.”
He also defended his move to get US troops out as part of his wider effort to bring Americans home from "endless wars," despite being excoriated by members of his own Republican Party.
US officials say, however, that those troops were expected to be repositioned in the region. Some of them could go to Iraq.
"I viewed the situation on the Turkish border with Syria to be for the United States strategically brilliant," Trump said.
"Syria may have some help with Russia, and that's fine. It's a lot of sand," he later said. "So you have Syria and you have Turkey. They're going to argue it out, maybe they're going to fight it out. But our men aren't going to get killed over it."
Acting last week after a phone call on October 6 with Turkish President Tayyip Erdogan, Trump abruptly upended five years of US policy with his decision to withhold protection from Syria's Kurds and to withdraw first about 50 special operations forces and then the roughly 1000 US troops in northern Syria.
"This is a mistake worse than what (Barack) Obama did" when the former president withdrew US troops from Iraq in 2011, Republican US Senator Lindsay Graham, usually among Trump's strongest supporters, told reporters.
The White House, fighting the domestic political damage, released a letter dated October 9 by Trump to Erdogan that said: "Don't be a tough guy" and "Don't be a fool!", urging him to halt the offensive.
Washington announced sanctions on Monday to punish Turkey, but Trump's critics said the steps, mainly a steel tariffs hike and a pause in trade talks, were too feeble to have an impact.
On Wednesday, US Treasury Secretary Steven Mnuchin said sanctions included the entire ministries of energy and defence and could be broadened to others. Republicans in the US House of Representatives plan to introduce sanctions legislation.
Trump dispatched some of his top aides, including Vice President Mike Pence and Secretary of State Mike Pompeo to Turkey for emergency talks to try to persuade Ankara to halt its assault. Trump said he thought Pence and Erdogan would have a "successful meeting" with Erdogan, adding that if they did not, US sanctions and tariffs "will be devastating to Turkey's economy."
Erdogan's spokesman said Turkey's Foreign Ministry was preparing retaliation for US sanctions.
A meeting between Trump and Republican and Democratic lawmakers was cut short - US House Speaker Nancy Pelosi, the top congressional Democrat, said Trump had a "meltdown" over a 354-60 House vote condemning his Syria withdrawal. Republicans said Pelosi "stormed out."
Dozens of Trump's fellow Republicans joined the majority Democrats in the vote. The split underscored deep unhappiness in Congress over Trump's action, which many lawmakers view as abandoning Kurdish fighters who had been loyally fighting alongside Americans to defeat Islamic State.
Trump has denied giving a green light to Turkey to attack the Syrian Kurds.
Erdogan has insisted there will be no ceasefire, and said he might call off a visit to the United States in November because of the "very big disrespect" shown by US politicians.
He also denounced Washington for taking the "unlawful, ugly step" of imposing criminal charges against a Turkish state bank over charges it broke sanctions on Iran. Washington says the case is unrelated to politics.
Turkey's assault has spawned a humanitarian crisis, with 160,000 civilians taking flight, a security alert over thousands of Islamic State fighters abandoned in Kurdish jails, and a political maelstrom at home for Trump.
Syrian government forces, backed by Washington's adversaries Russia and Iran in the more than eight-year civil war, have taken advantage of the power vacuum left by US troops to advance into the largest swath of territory previously outside their grasp.