Since the Russian invasion in February, he has not travelled outside of the country.
Unnamed authorities told US media that Mr. Zelensky could also address Congress. Neither the visit nor the plans have received formal confirmation.
Security concerns would make a visit risky, so last-minute changes to plans are a possibility.
Democratic House Speaker Nancy Pelosi asked members of Congress to attend the event on Wednesday night in a letter on Tuesday.
She said in the letter: “Please be present for a very special focus on democracy.”
Mr. Zelensky has visited troops all around Ukraine and frequently receives foreign officials in the country’s capital, Kyiv. On Tuesday, he paid an unexpected visit to the battleground city of Bakhmut.
The president of Ukraine has frequently communicated with foreign leaders via phone and video call, often from his office in Kyiv.
But an unannounced trip abroad would be a first since the war started, and it would also underscore the significance of Ukraine’s relationship with the US, which has been instrumental in providing military support.
Since the beginning of the conflict, the US has invested $18.5 billion (£15.1 billion) in direct military support, significantly more than any other nation.
The monthly cost of defence for Ukraine, according to President Zelensky, is about $5 billion (£4.1 billion). He has pleaded for more financing.
The administration of President Joe Biden is reportedly considering giving Ukraine Patriot missiles so that it can defend its crucial infrastructure from Russian strikes, according to recent reports in US media.
Russia has been attacking Ukraine’s energy industry, leaving millions in the dark during winter months when the temperature is well below zero.
President Zelensky unexpectedly paid a visit to the front-line city of Bakhmut on Tuesday, where Ukrainian and Russian soldiers have been engaged in a bloody, months-long conflict.
According to the president, he visited with soldiers and presented honours to military.
The visit was a powerful act of resistance and a show of solidarity with Ukrainian forces fighting in some of the bloodiest engagements in recent weeks.