The US meeting - the first since Trump took office - will be a key chance for Erdogan to build a strong alliance with Trump as Ankara's relationship with the European Union goes through one of its worst ever crises.
"The date has been fixed," Erdogan told A News in an interview on Thursday, saying the trip would happen after he concludes a visit to China on May 15.
"I hope and pray that this tete-a-tete meeting in the United States in May will lay the foundation for a stronger cooperation."
The announcement came after Trump surprised many observers by phoning the Turkish leader to congratulate him for winning a controversial weekend referendum on enhancing his powers.
Russian President Vladimir Putin also called his Turkish counterpart to offer congratulations.
The 'Yes' camp won Sunday's poll with just 51.41 percent of the vote but the result has been challenged, with the opposition claiming the vote was rigged and angry protests erupting in parts of Istanbul.
The European Union has also urged a probe into fraud claims after international observers voiced concerns.
In the last few months of Barack Obama's presidency, relations between the NATO allies hit a rough patch with Turkey furious over US backing for a Kurdish militia in Syria it sees as a terror group.
Ankara is also pressing Washington to extradite Fethullah Gulen, the US-based Islamic preacher it accuses of being behind last year's failed coup. Gulen denies the charges.
Asked about his potential future relationship with Trump, Erdogan said: "It's very early to comment. When I met president Obama for first time I was very hopeful. Let's embark on this journey."
But he appeared upbeat about the prospects, saying the two had already spoken three times by phone. "Every time we spoke to one another, I have become more hopeful."
He also said that Trump's background as a successful businessman was a help.
"We have an advantage. President Trump is a very successful businessman, he is tycoon in real estate, he was a brand."
He also expressed confidence that Washington would extradite Gulen, saying: "I believe president Trump will do what president Obama failed to do."
Indicating that major meetings with EU leaders were not currently on the agenda, Erdogan said his schedule for trips after the referendum also included Russia and India.
Turkish media said a meeting with Putin would be held on May 3.
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