Polish President Andrzej Duda went as far as to suggest the new military base could be named “Fort Trump”.
Speaking during a joint news conference at the White House, Mr Duda said: “I invite you to post more American military troops in Poland.”
The Polish commander-in-chief said US presence in his nation would be a “guarantor of security”.
Mr Duda revealed he thought it could be “possible” to build a US base in Poland and that it would serve in both US and Polish interests.
To which Mr Trump disclosed he was considering Warsaw’s request but added Poland “would pay the United States”.
He said: “He would pay the United States, meaning Poland would be paying billions of dollars for a base.
“We’re looking at that more and more from the standpoint of defending really wealthy countries.
“The President offered us much more than $2billion (£1.5billion).”
The proposed military base concurs with Poland’s desire to strengthen its defences against an increasingly bellicose Russia.
Trump shared Mr Duda’s concern during the White House conference and claimed Russia has “acted aggressively”.
He said: “I think it’s a very aggressive situation.
“I think that Russia has acted aggressively. They respect force.
“They respect strength, as anyone does.
«And we have the greatest strength in the world, especially now.”
Poland signed up to NATO in 1999 despite Russia’s disagreement.
The US currently supplies the majority of the troops, about 900, for a NATO battalion in Poland as part of the alliance response to Russian intervention in Ukraine.
The troops are also in response to Moscow’s muscle-flexing along its borders with the West, including large-scale military exercises like one held with China last week.
Poland has long argued for increased US military presence and a permanent US base on Polish land.
Mr Duda said: “We believe that the presence of the United States is a guarantor of security in our part of Europe.
“I was smiling when I was talking to Mr President. I said that I would very much like for us to set up a permanent American base in Poland, which we would call Fort Trump.
“I firmly believe that this is possible. I am convinced that such a decision lies both in the Polish interest and in the interest of the United States.”