Earlier on Friday, Washington rejected Moscow's proposal to unconditionally extend New START, which expires in February, for one year and continue negotiations on the new pact to replace the only arms control regime between the two countries.
"We are in fairly close contact with our USA colleagues on a range of issues related to strategic stability, where we would like to underline the continuing strength and growing urgency of our initiative to decide as soon as possible on the extension of the current New START Treaty, which is about to expire in February next year, for the next five years without any preconditions", Russian Foreign Minister Sergei Lavrov said at the meeting.
New START limits each country to no more than 1,550 deployed nuclear warheads and 700 deployed missiles and bombers.
He followed up his remarks with a post on Twitter in which he made clear the USA will not accept the new Russian offer.
Incidentally, a potential arms race was what Putin was warning against as he extended his offer to the US.
O'Brien reiterated the United States proposal of an extension of New START for one year, during which both countries cap all nuclear warheads, referring to both strategic and tactical ones.
A failure to extend the pact would remove all constraints on USA and Russian deployments of strategic nuclear weapons and their delivery systems, fueling a post-Cold War arms race and tensions between Moscow and Washington.
Lavrov said Moscow favoured extending the current treaty for another five years without any conditions but was also ready to work out a new agreement with the Americans. "Yet, we are open to discussing this issue as well", Putin said on Friday.
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US Secretary of State Mike Pompeo recently suggested that Washington intends to seek an early agreement on the treaty.
On Tuesday, Ryabkov said: "The US position in favor of freezing has always been known to us, it is unacceptable to us".
This would have been a win for both sides, he said. The US wants other weapons to be included in a new arms control deal.
He said while the US wants a cap on all USA weapons in exchange for a New START extension, it remains an open question what it is willing to give the Russians in return.
Russia's arsenal of short-range tactical nuclear weapons is believed to be bigger and more varied than that of the United States so it would be a "one-sided concession" on Moscow's part, he said.
But time is running out, he added.
Earlier this week, Mr Lavrov voiced scepticism about reaching a deal on New Start, noting that Russian Federation can not accept the conditions put forward by the United States for its extension.
Jacqueline Feldscher and The Associated Press contributed reporting.