Since the first American shipments of sophisticated weapons to Ukraine, President Biden has never wavered on one prohibition: President Volodymyr Zelensky had to agree to never fire them into Russian territory, insisting that would violate Mr. Biden’s mandate to “avoid World War III.”

But the consensus around that policy is fraying. Propelled by the State Department, there is now a vigorous debate inside the administration over relaxing the ban to allow the Ukrainians to hit missile and artillery launch sites just over the border in Russia — targets that Mr. Zelensky says have enabled Moscow’s recent territorial gains.

The proposal, pressed by Secretary of State Antony J. Blinken after a sobering visit to Kyiv last week, is still in the formative stages, and it is not clear how many of his colleagues among Mr. Biden’s inner circle have signed on. It has not yet been formally presented to the president, who has traditionally been the most cautious, officials said.

The State Department spokesman, Matthew A. Miller, declined to comment on the internal deliberations over Ukraine policy, including Mr. Blinken’s report after his return from Kyiv.

But officials involved in the deliberations said Mr. Blinken’s position had changed because the Russians had opened a new front in the war, with devastating results. Moscow’s forces have placed weapons right across the border from northeastern Ukraine, and aimed them at Kharkiv — knowing the Ukrainians would only be able to use non-American drones and other weaponry to target them in response.

For months, Mr. Zelensky has been mounting attacks on Russian ships, oil facilities and electricity plants, but he has been doing so largely with Ukrainian-made drones, which don’t pack the power and speed of the American weapons. And increasingly, the Russians are shooting down the Ukrainian drones and missiles or sending them astray, thanks to improved electronic warfare techniques.