Polish President Andrzej Duda told reporters after meeting with Trump at the White House on Tuesday that he quickly dubbed the base "Fort Trump", an obvious appeal to the businessman turned US president who has hotels, condos and golf courses around the globe named after himself.
"If they're willing to do that, it's something we will certainly talk about", he added.
In March, the United States and Poland said technical problems would delay completion of the Polish section of a U.S. missile defense shield by two years to 2020.
Speaking at a joint press conference ending Mr Duda's visit to Washington DC, Mr Duda said: "I invite you to post more American military troops in Poland". USA soldiers are now present on its soil through NATO's back-to-back rotation.
Still, Poland's offer falls far short of replacing base arrangement the United States has in Germany, home to the massive Ramstein Air Base and major U.S. Army training grounds in the Bavarian towns of Grafenwoehr and Hohenfels. At one point earlier this year, in a meeting with his advisers, Trump inquired about American troops stationed in Germany and asked about possibly removing them. We want to implement more projects.
"I am with the president".
Poland has repeatedly requested a permanent USA military presence on its soil.
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"I think its a very aggressive situation".
Mr Duda's latest plea comes after Warsaw's defence ministry offered the USA up to £1.52billion ($2bn) to increase the United States presence in the country as a deterrent against a potential Russian aggression. I think that Russian has acted aggressively.
"The United States commits to explore options for an increased USA military role in Poland and we will intensify our consultations to determine the concept's feasibility", the White House said in a statement later.
"They respect force. They respect strength as anyone does".
President Donald Trump is welcoming the leader of Poland to the White House.
U.S. President Donald Trump speaks next to first lady Melania Trump during a meeting with Poland's President Andrzej Duda (L) in the Oval Office of the White House in Washington, U.S., September 18, 2018.
Poland, which is a member of North Atlantic Treaty Organisation since 1999, has been taking a very active part in the recent military activities of the alliance, described as an attempt to counter "hostile" Russian moves.
Warsaw was the first stop on Trump's first European tour past year.