Fox News correspondent Peter Doocy asked Mr Biden what he discussed with Mr Putin on a recent call between the leaders.
The president replied “You. He sends his best” as he walked away from the press.
White House Press Secretary Jen Psaki confirmed that Mr Biden called Mr Putin on Tuesday.
“President Biden made clear that the United States will act firmly in defense of its national interests in response to actions by Russia that harm us or our allies,” the White House said in a statement. “The two presidents agreed to maintain transparent and consistent communication going forward.”
According to the White House, Mr Biden told Mr Putin that the US would defend itself from “malign actions by Russia”.
“His intention was also to make clear that the United States will act firmly in defense of our national interests in response to malign actions by Russia,” Ms Psaki said.
The White House said part of the discussion included a five-year extension to the New START nuclear arms reduction treaty signed between the countries in 2010.
Ms Psaki said Mr Biden also pressured Mr Putin on Russia’s “ongoing aggression” toward Ukraine and addressed reports that Russia was offering bounties on US military members in Afghanistan. That story broke during Donald Trump’s administration, and Mr Trump did not address the issue with Mr Putin.
Mr Biden also reportedly discussed Russia’s alleged interference in the 2020 US election and Mr Putin’s ongoing attacks against the country’s opposition leader, Alexei Navalny. Mr Navalny was nearly killed last year when he was poisoned on an aeroplane.
The president also reportedly discussed a massive cyberattack last year that targeted numerous US government agencies and private companies. Both US intelligence officials and private cyber security firms have pointed to Russia as the culprits of the hack.
Mr Navalny accused Mr Putin of being behind the poisoning. Protesters supporting Mr Navalny who opposed his arrest in Russia have been met with violence by the country’s police.
The Kremlin had a different account of the call, stating that Mr Putin and Mr Biden discussed “topical issues” like the economy and the coronavirus. It did mention that the “Ukraine domestic settlement” was discussed, but did not offer further detail.
While Mr Biden made the call, sources speaking with CNN said the Kremlin requested the correspondence.
Mr Biden told reporters Monday that he hoped to discuss troubling issues with Russia while still leaving the potential for further diplomacy and normal relations on the table.
“I find that we can both operate in the mutual self-interest of our countries, as a New START agreement, and make it clear to Russia that we are very concerned about their behaviour, whether it’s Navalny, whether it’s the Solar Winds or whether it’s reports of bounties on the heads of Americans in Afghanistan,” he said.
Mr Biden reportedly wanted to call other European leaders prior to his call with Mr Putin to gauge their concerns.
The president received briefings from senior administration officials prior to his call with Mr Putin, a break from Mr Trump’s habit of entering calls without preparation.
Mr Putin was one of the last world leaders to congratulate Mr Biden on his electoral victory. The Russian president waited until the electoral votes were counted on 6 January to send his congratulations.