By Agence France-Presse
Ankara's push to buy Moscow's S-400 missile defence system has strained ties between the NATO allies, with the threat of penalities from Washington looming over Turkey.
"We heard from him that there won't be anything like this (sanctions)," Erdogan told a press conference, after meeting with the US president on the sidelines of the summit in Osaka, Japan.
While Erdogan insisted Turkey and the United States were "strategic partners", he said that "no one has the power to intervene in Turkey's sovereignty".
His office said Trump wished to resolve the S-400 issue "without damaging bilateral ties".
Before the talks, Trump said Turkey "has been a friend of ours... We're a big trading partner. We're going to be much bigger."
Despite heavy pressure from Washington to cancel the purchase, Erdogan has repeatedly said it was a "done deal" and reaffirmed on Saturday that delivery of the system would begin in the first half of July.
Experts say sanctions would hit Ankara's already fragile economy hard. Tariffs imposed by Trump last summer over the jailing of a US pastor helped trigger a currency crisis.
Washington has warned that if the S-400 system is delivered to Turkey, the country faces penalties under the Countering America's Adversaries Through Sanctions Act (CAATSA) which bars business with Russia's state and private defence and intelligence sectors.
Turkish officials have previously said they are preparing for US sanctions.
In a line of argument that appears to have played well with Trump, Erdogan told him that the reason for purchasing the S-400 was that his predecessor Barack Obama had failed to secure a deal to sell Turkey the American Patriot system instead, with the sale blocked by Congress at the time.
The Patriot is an anti-missile and anti-aircraft weapon system, similar to the S-400. The US finally approved the sale of the Patriot system to Turkey in December.
Trump said Erdogan should not be blamed for Obama's failure.
"We have a complicated situation because the president (Erdogan) was not allowed to buy the Patriot missiles... he wasn't allowed by the Obama administration," Trump told reporters in Osaka.
"So he buys the other missile and then, all of a sudden, they say, 'Well, you can now buy our missile,'" Trump said, adding: "You can't do business that way. It's not good."
Washington has nonetheless threatened to remove Turkey from its F-35 fighter jet programme, giving Ankara until July 31 to cancel the S-400 purchase or have its pilots kicked off the training course and expelled from the US.
Turkey has plans to buy 116 F-35s, Erdogan said, and has invested a total of $1.4 billion in the production so far.
Relations between Turkey and the US have been tense over multiple issues, including American support for a Kurdish militia in Syria and the failure to extradite a Pennsylvania-based Muslim preacher blamed for the 2016 failed coup.
But the two leaders said they were committed to increasing bilateral trade.