"And I firmly believe that this is possible", he added. "I am convinced that such a decision lies both in the Polish interest and in the interest of the United States".
The president raised his eyebrow at the idea but didn't rule it out. "The presence of the United States is a guarantor of security in our part of Europe". "If they're willing to do that (pay), it's something we will certainly talk about".
Warsaw could offer up to US$2 billion in funding for a U.S. military base, according to a Polish Defense Ministry proposal reported by Polish media.
A senior Polish government official said Warsaw would potentially be willing to spend even more over time or offer additional incentives to Washington. "I want to commend Poland for meeting its North Atlantic Treaty Organisation defense spending obligations and I am glad that it plans to increase spending beyond the 2 percent minimum obligation".
The United States now rotates troops through Poland temporarily but permanently stationing forces there would be expensive given costs that can include housing for families, schools and hospitals. "And we're very close to being finished with that job", he said.
Some allies such as Germany, however, have argued a permanent North Atlantic Treaty Organisation force in Poland or the Baltics would only escalate tensions with Moscow.
Poland has long sought to have the USA bolster its presence in Eastern Europe as a counter to Russian Federation, and Polish President Andrzej Duda said at the White House that the base could be called "Fort Trump" if that would speed the process.
Germany has always been the primary base for American forces in Europe, with approximately 35,000 USA service members stationed in the country.
A decision from the United States could come early next year.
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The Pentagon, however, has said the United States remains fully committed to Germany.
Trump - accused by political opponents of having colluded in a shadowy Russian operation to aid his surprise 2016 election win against Democrat Hillary Clinton - agreed with Duda's assessment.
The comments came at a joint White House press conference by the leaders Tuesday. Equipment of the previous USA unit stationed there, has, however, "remained in place" for reasons unexplained, according to Russian military.
Mr Trump, who seemed enticed by the offer, said he deemed the security of Poland "almost as important" to him as it is to the Polish president.
At a press conference afterward, Duda said Poland would name the facility after Trump.
"It is the reality that we live in today", Duda said.
Trump (R) and Duda walk to the Oval Office.
"I feel so badly for him that he's going through this, to be honest with you, I feel so badly for him", Trump said.
Poland and other Eastern European countries have been lobbying the United States to increase the number of troops on the continent as they worry about combating an increasingly aggressive Russian Federation.
"They respect force. They respect strength as anyone does".