The visit came at the end of last week, The Washington Post first reported.
In previous visits to the region before the war, Mr Burns met Russian President Vladimir Putin in an attempt to dissuade him from invading Ukraine. The CIA director had also accurately warned Ukrainian officials of Russia's plan to attack a Ukrainian airport in the early stages of the war.
In their meeting last week, Mr Zelenskyy pressed Mr Burns on how long assistance from Washington would last under a Republican-led House of Representatives, the Post reported.
Kyiv believes the $45 billion the US allocated to fund Ukraine could last through the summer, but is watching to see whether Congress passes another funding package as it did last year.
Before Republicans won the House in the 2022 midterm elections, Representative Kevin McCarthy, now House Speaker, said his party would be less willing to write a “blank cheque” for Ukraine upon taking control.
That would pit him against Senate Republican leader Mitch McConnell, who has urged the White House to increase its assistance to Kyiv.
President Joe Biden's administration on Thursday evening announced an additional $2.5 billion security package for Ukraine as the war drags into its 11th month.
The latest package includes 90 Stryker armoured personnel carriers, 59 Bradley Infantry Fighting Vehicles, eight Avenger air defence systems and ammunition, the Pentagon said.
Meanwhile, Defence Secretary Lloyd Austin and Chairman of the Joint Chiefs of Staff Gen Mark Milley were at the US airbase in Ramstein, Germany, to discuss providing tanks to Ukraine with other defence leaders.
Germany has so far refused to deliver Leopard 2 tanks to Ukraine despite pressure from western allies.