The move comes on the heels of a declassified United States intelligence report that claimed President Putin had authorised "influence operations" to undermine then-Democratic candidate Biden during the 2020 election.
"These are really bad remarks by the USA president".
Putin responded Thursday by saying, "It takes one to know one".
Mr Biden was quick to extend a key nuclear arms pact with Russian Federation after he took office in January.
Jen Psaki, White House spokesperson, added: "Certainly the Russians will be held accountable for the action they've taken".
"The president believes that one of the greatest attributes of the United States is our honest self-reflection and our constant striving for progress, and there's always more work to do", she said.
Suggesting Biden was hypocritical in his remarks, Putin said that every state had to contend with "bloody events" and added Biden was accusing the Russian leader of something he was guilty of himself.
Putin said Russian Federation would continue to work with the U.S.in certain areas, but his spokesman said earlier that Mr. Biden's remarks were evidence the new American administration "does not want to get the relationship with our country back on track".
"I would say to him: 'Be well.' I wish him good health". He wouldn't answer if Russian Federation could go as far as to rupture diplomatic ties with the United States.
Russia wants an apology from U.S. after Biden called Putin a killer, says Kremlin ally
He said it would be an "open direct discussion" that would be "interesting" for the people of Russian Federation and the US. "And they will have to reckon with that".
Russia's President Vladimir Putin chairs a joint meeting of the country's State Council and the Council for Strategic Development and National Projects via a video conference call at the Novo-Ogaryovo state residence outside Moscow, Russia, December 23, 2020.
The exchange of tough statements comes on the heels of a declassified report from the U.S. national intelligence director's office that finds President Vladimir Putin authorized influence operations to help Donald Trump in last November's presidential election.
The reaction to Biden's comments highlighted the tense relations between Russia and the USA that appear likely only to worsen in the coming weeks as the Biden administration looks set to respond over a variety of alleged Russian misdeeds.
A Google translation of the tweet reads: "Putin invited Biden to talk live: In order not to dive in absentia, I want to propose to continue our discussions".
He also pointed out that people usually project onto others what they think about themselves.
Other Russian officials and lawmakers were less diplomatic.
Upon entering office, Biden complained that the Trump administration had not left much infrastructure for the ambitious vaccination program, but he still aimed to deliver 100 million doses - enough for almost one-third of the USA population - by his 100th day in office. But his administration has said it will take a tougher line with Moscow than Washington did during former US President Donald Trump's term in office, and engage only when there is a tangible benefit for the United States. The Russian authorities rejected the accusations.
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