An additional 1,000 USA military troops will be deployed to Poland to join about 4,500 now there, a senior White House official said.
"We haven't totally made up the decision", he told reporters as he appeared with Duda in the Oval Office.
Mr Duda suggested he might name the installation "Fort Trump" after the US President.
"We are protecting Germany from Russian Federation and Russia is getting billions and billions (of dollars)", Trump said.
"It actually came pretty close to a halt over the White House", Trump said, about the F-35 flyby, commenting on its flight capabilities.
He told reporters: "I hope that Poland is going to have a great relationship with Russian Federation".
"It's. something that I've been looking at and I'm thinking about", he said. "I think because of what you've done, and the strength, and maybe we help also, because of what we're doing and doing for Poland".
However, the United States president once again stumped everyone by insisting he would like a "great relationship" with Russia, China "and many other countries".
USA officials said earlier this week that Trump, in addition to the additional troops, would send a squadron of Reaper drones to Poland to aid its self-defense amid concerns about Russian military activity.
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The additional USA troops will be stationed at various Polish bases in the country.
Establishment of an Air Force MQ-9 Reaper Intelligence, Surveillance and Reconnaissance squadron in Poland. Trump also voiced support for Polish plans to construct a Baltic pipeline and announced expanded co-operation between the allies on civilian nuclear energy.
"It is going to be an enduring [U.S. military] presence that will hopefully increase gradually", Duda said.
During meetings throughout the day, Trump had announced that the United States would deploy 1,000 soldiers to Poland as well as a squadron of MQ-9 Reaper drones. He offered no timetable for when the additional military personnel would begin arriving in Poland.
"You people would have a field day about that", Trump said from the White House during an appearance with Polish President Andrzej Duda, who first broached the idea of creating a Fort Trump in Poland during a previous sitdown. Duda looked up and waved as the jet passed. While the press conference focused on areas of agreement, Trump said he differed with Poland's efforts to force the country's Supreme Court justices to retire early.
"Everything is excellent", Duda added. Critics have accused the Polish government of taking steps to undermine the judiciary and the news media. The visit, Duda's second within a year, was ostensibly to deal with two issues: Poland's desire for an expanded USA military presence and the US's desire to sell natural gas to Poland. "This shows the strong commitment of the United States to European security & the strength of the transatlantic bond", he said in a tweet.
Likely responses would include the Kremlin increasing military assets in the Russian territory of Kaliningrad, which borders Poland, and possibly also in Belarus, a close ally of Moscow that has resisted hosting a full-scale Russian base on its territory.
"Germany is not living up to what they are supposed to be doing with respect to North Atlantic Treaty Organisation, and Poland is".
The Eastern European nations have reached out to the USA and North Atlantic Treaty Organisation for greater protection, fearing becoming the next target of Russia's military advance.