He said the two countries had reached a political agreement on licenses for the production of the missiles.
The Patriot is a US-made air defence system. Its PAC-3 interceptor - short for Patriot Advanced Capability-3 - is one of the few Western weapons capable of shooting down the ballistic missiles Russia has increasingly fired at Ukrainian cities.
Zelenskiy, speaking to reporters after returning from a NATO summit and talks with US President Donald Trump in Turkey, also said that talks were proceeding with the US on a "drone deal" or joint drone production.
"I believe this was a productive summit for Ukraine. In the coming days, we'll receive a package from the United States, and there were also some separate agreements," Zelenskiy said, referring to securing the PAC-3 interceptors.
"We resolved this issue politically," he said of producing Patriot missiles for Ukraine. "It's now very important that our technical teams, all our representatives from different ministries, representatives of the executive branch, start working on this without delay, so that we can get licences very quickly and start production in Ukraine as soon as possible."
Zelenskiy has long pleaded for faster supplies of interceptors capable of bringing down Russian ballistic missiles, a need accentuated by recent assaults on the Ukrainian capital and other cities.
He said no drone deal had yet been signed with Washington, "but there are some documents that have already been signed so that the American side can get from Ukraine various types, different, different kinds that the United States is interested in for testing. And they're getting them from us."
He said these included "both aerial drones and marine drones, as well as other technological things".
He also said discussions would proceed with Ukraine's European allies on developing a separate anti-missile system, with a meeting planned for France in the near future.
The Ukrainian president has improved his relations with Trump through a series of meetings after an encounter last year in the Oval Office descended into a shouting match.
Zelenskiy told reporters Trump was now "positive about Ukraine."
"We definitely discussed a great many different things. We need to follow up, prepare for diplomacy, and there are a few other things, but all of this is the right direction," he said. "During the meeting he was very constructive."
US-brokered talks aimed at moving toward a peaceful resolution with Russia have stalled, with Washington focused on the conflict in Iran.
Trump said this week that he and Zelenskiy had developed a "very good" relationship and both Moscow and Kyiv wanted to end the war that began with Russia's invasion in February 2022.
Trump spoke to both Zelenskiy and Russian President Vladimir Putin ahead of the summit. Putin has said he will press ahead with his war despite mounting difficulties like fuel shortages linked to Ukrainian drone strikes on oil industry targets.
Moscow has demanded that Kyiv cede the rest of its eastern Donetsk region that it has been unable to conquer in more than four years of fighting.
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