Washington has said that if fellow North Atlantic Treaty Organisation member Turkey does not cancel the S-400 deal by July 31, Ankara will be blocked from purchasing the next-generation F-35 fighter.
"As far as I understand, there are no delays in the process of this delivery", Erdogan said during his meeting with Russian President Vladimir Putin on the sidelines of the G20 summit in Japan's Osaka.
The presidency added Trump told the Turkish leader he hoped the S-400 issue would be solved "without damaging bilateral relations".
The NATO allies have been at odds over Turkey's decision to procure the Russian S-400 missile defence systems, with Washington warning of sanctions if the deal goes through.
Under possible US sanctions, Turkey could face expulsion from the F-35 programme, a move Erdogan has dismissed. "We're going to be much bigger".
Erdogan also said that he expected the S-400 delivery to start in the first half of July, adding that Turkey had invested $1.4 billion (€1.23 billion) in its production so far.
The two leaders met for about 35 minutes.
Ankara has repeatedly stated that it will stick with the S-400 deal, even if it means forfeiting its participation in Washington's F-35 program.
In an effort to sway Turkey, the United States has offered to supply it with Raytheon Co Patriot missiles, but Erdogan has said the USA offer was not as good as Russia's S-400 proposal. Turkey has been a friend of ours...
Trump noted that Turkey was barred from purchasing Patriot systems during the Obama administration, but stressed that his administration has a difference stance on the matter. "He made a deal, he paid them a lot of money, put up a lot of money. God willing, the system will be delivered to our country next month". "He got treated very unfairly".
To foster better ties with Ankara, Trump promised to visit Turkey "soon", but did not specify any dates.