His remarks came after Mr Zelensky called on Britain, the US and France to lift arms restrictions and allow Ukraine to strike deep into Russia.
In an interview with The New York Times, he said that he would raise the demand personally with Sir Keir, Mr Biden and Emmanuel Macron.
All three must sign off on the use of Storm Shadow missiles against targets in Russia.
Mr Macron has already publicly said he would agree to the move, while Sir Keir has privately indicated he is willing to.
The missile was jointly developed by Britain and France but relies on a US targeting system, giving Mr Biden a veto.
World leaders will meet in New York against the backdrop of Russia making some of its swiftest gains in the war in more than two years.
Moscow’s forces have been using a “scorched earth” tactic to capture Vuhledar, a coal-mining town in a key strategic location.
It has never before been captured, but pro-Kremlin military bloggers appeared confident that Kremlin forces would overwhelm the Ukrainian defenders.