The Oct 9 letter was released by the White House on Wednesday as the US president 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 Oct 6 phone call.
"Let's work out a good deal!" Trump said in the letter. "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."
The BBC has said that the Turkish president's office "rejected" the missive.
Mark Lowen, the BBC's Turkey correspondent tweeted: "When President Erdogan received Trump's letter, he 'thoroughly rejected it, threw it in the bin' and launched the Syria operation the same day.
"His mind was already made up to move in; letter was reportedly the final straw."
Trump had the letter released to bolster his view that he did not give Turkey a green light to invade Syria. Many US 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 a copy of a letter Gen Mazloum had sent him.
"History will look upon you favorably 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."
On October 9, Turkish military forces and Ankara-backed militants launched a long-threatened cross-border invasion of northeast Syria in a declared attempt to push Kurdish militants from the People's Protection Units (YPG) away from border areas.
Ankara views the US-backed YPG as a terrorist organization tied to the homegrown Kurdistan Workers' Party (PKK), which has been seeking an autonomous Kurdish region in Turkey since 1984. The YPG constitutes the backbone of the SDF.