President Trump is meeting with Poland's President Andrzej Duda at the White House Tuesday.
"I invite you to post more American military troops in Poland", Duda said, calling a United States presence a "guarantor of security".
At a press conference afterward, Duda said Poland would name the facility after Trump.
In his remarks, Duda said, "I would very much like to ask to set up a permanent American base in Poland, which we would call 'Fort Trump.' I firmly believe this is possible".
Statements made by the two presidents come after Poland has been requesting an increased USA presence in Poland in light of increased Russian military activity.
The two leaders also agreed to collaborate on energy, although no specific deal was announced.
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Trump says that he's considering the idea and that Poland has offered more than $2 billion to the U.S.to pay for such an effort.
Poland has repeatedly requested a permanent United States military presence on its soil. "And we have the greatest strength in the world, especially now", Trump said. "I am convinced that such a decision lies both in the Polish interest and in the interest of the United States".
"The presence of the USA armed forces in this area is absolutely justified", Duda said. Warsaw, long uneasy about a more aggressive Russia, has argued an Army brigade and associated support units stationed permanently in the country would serve as a more serious deterrent to Russian aggression. I think that Russian has acted aggressively.
Security, trade and energy topped the agenda as Trump welcomed Duda to the White House for the first time.
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. Another concern is setting up a permanent base in Poland would cause a rift among North Atlantic Treaty Organisation members and chip away at alliance unity.
Trump is reported to have ordered a review of costs of basing United States troops in Germany, against a backdrop of poor relations with Germany. Since taking office, the Polish government has unnerved Brussels with policies seen as restrictive of the courts and the news media, and is now in direct conflict with the European Union over its courts.