Although he himself is a former Communist, Mr Kwasniewski has successfully reinvented himself as a Western-style social democrat and was quick to hail Mr Yeltsin's victory over his Communist rival Gennady Zyuganov as good news for Poland and its prospects for joining the military alliance.
In a series of meetings with President Bill Clinton and other senior American officials, he will call for decisive action on the issue now that the victory of Mr Yeltsin - who is slightly less opposed to Nato enlargment than Mr Zyuganov - appears to have opened a window of opportunity. "For a long time we have been asked to be patient because of the Russian election," said Marek Siwiec, a presidential adviser in Warsaw. "Now . . . we want to know what the real timetable for enlargement will be."
Since defeating Lech Walesa in last November's presidential election, Mr Kwasniewski has sought to reassure Western leaders that Poland remains committed to joining both Nato and the European Union and that it could make a positive contribution to both. "This [US] visit should show that governments can change, presidents can change, people can change, but [Polish] policy does not change," he said.
It is an open secret that Poland is among the front-runners from central and east Europe for Nato membership, possibly as early as 1999. The country's cause has recently received a boost in the form of an unexpectedly strong endorsement from Bob Dole, Mr Clinton's Republican rival in the November US presidential election. Accusing Mr Clinton of dragging his feet on enlargement, Mr Dole has singled out Poland, Hungary and the Czech Republic as all being ready for Nato membership now and called for their speedier integration.
Polish officials privately doubt whether a President Dole would bring them into the alliance any sooner. But they are glad that the issue is being aired in the US. "The more discussion on the subject the better," a Foreign Ministry source said. "We want people to see that Poland is not some sort of desert ... there are military installations here, we can fulfil military missions (in Bosnia for example) and bringing us into Nato won't be as expensive as some fear."
While in Washington, Mr Kwasniewski will meet the Defense Secretary, William Perry, to discuss Poland's planned purchase of a fleet of jet fighters, possibly from Lockheed Martin or McDonnell Douglas, in a deal which could be worth some $3-5bn (pounds 1.9-3.2bn).
He is also planning a series of meetings with US Jewish leaders to discuss a $100m plan to turn the former Nazi concentration camp at Auschwitz into a fully protected zone and a centre of learning.
Polish-Jewish relations plummeted earlier this year when it emerged that a local developer wanted to build a supermarket and fast-food outlet right next to the Auschwitz camp gates. Mr Kwasniewski, quickly condemned the supermarket proposal, thereby helping to ensure it was blocked.Reuse content