In the letter dated 9 October, and sent after US troops were pulled out of Syria, Mr Trump told Mr Erdogan: "Don’t be a tough guy. Don’t be a fool!"
President Trump was urging Turkey not to launch a military offensive against Kurdish-led forces in northern Syria, but Mr Erdogan ignored this request.
US Vice President Mike Pence is now in Ankara to push for a ceasefire.
The US has faced intense criticism for the withdrawal of troops, which critics say gave Turkey the green light to launch the military attack.
In recent years, the Kurdish-led Syrian Democratic Forces (SDF) alliance has been a critical ally to the US in the fight against the Islamic State (IS) group in Syria.
There are fears that the destabilisation of northern Syria could lead to a jihadist resurgence.
