American officials believe the calculus over both missiles has changed, after Iran shipped long-range weapons to Russia.
Asked about the issue on Tuesday, Mr Biden confirmed: “We are working that out right now.”
Antony Blinken, the US Secretary of State, said on a visit to Kyiv on Wednesday that he had discussed the missiles and would “take that discussion back to Washington to brief the president”.
However, intelligence sources told the New York Times that Mr Biden is unlikely to give permission for Ukraine to use both Storm Shadows and ATACMS yet, amid concerns about escalation by Mr Putin. After the Storm Shadows approval, a further decision on ATACMS could take weeks.
American intelligence suggests that allowing American-made missiles to be fired on Russian targets could prompt the Kremlin to step up its support for Iran and its proxies in the Middle East, which are engaged in an ongoing conflict with Israel.
It could also escalate the conflict in Ukraine, where both sides are locked in battle over territory in the eastern Donbas region.
On Thursday, Mr Putin said allowing Western weapons to be used inside Russia would bring Nato into direct conflict with Russia.
“We are not talking about allowing or not allowing the Ukrainian regime to strike Russia with these weapons,” he said.
“We are talking about deciding whether NATO countries are directly involved in the military conflict or not.”
The use of long-range missiles is the latest debate over Western support for Ukraine, following similar discussions over the supply of HIMARS artillery, M1 Abrams tanks and F-16 fighter jets.
While the UK has been willing to allow Storm Shadows to be used in Russia for some weeks, the decision has been held up by Mr Biden’s resistance to Ukraine using American targeting systems alongside the missiles.
Germany, which has a large supply of similar Taurus missiles, has declined calls for it to send its own supplies, which defence sources believe would significantly boost Ukraine’s warfighting ability.
Ben Wallace, the former defence secretary, told The Telegraph long-range missiles would allow Ukraine to “layer its defence” to cover Russian attacks between 30km and 300km from the front line.
“It is a mystery to me why countries that say they want Ukraine to win, such as Germany and America, often end up in this tug of war for capabilities,” he said.
“We all know where this will end, and it’s best just to get on with it.
“Delay always benefits Russia. If we are serious about Ukraine we must make sure we remove from Russia’s grasp every factor that helps Putin now.”