In the closing days of his reelection bid, Trump has looked for ways to boost his foreign policy record, and although he says he favors nuclear arms control, he has called New START flawed and unfavorable to the U.S. Last year he withdrew the United States from a separate nuclear arms treaty with Russia, and he waited until this year to begin engaging the Russians on the future of the New START deal.
However, soon after the proposal was voiced, US National Security Adviser Robert O'Brien said that without freezing nuclear warheads such an extension would be a non-starter.
Tensions have raged for months over the fate of New START, which caps the number of nuclear warheads held by Washington and Moscow and expires on February 5.
The U.S. has insisted on broadening the terms of the 2010 treaty but hasn't been able to win Russia's support. "We hope that Russian Federation will re-evaluate its position before a costly arms race ensues", he said.
Moscow opposes a proposal by Washington for a new framework that would cover tactical nuclear weapons in addition to strategic warheads.
A failure to extend the pact would remove all constraints on USA and Russian deployments of strategic nuclear weapons and their delivery systems, fuelling a post-Cold War arms race and tensions between Moscow and Washington.
"For all these years the treaty has worked well, fulfilled its role as a limit on the arms race", Putin told a televised meeting of his Security Council on Friday.
We thought we made a really fair, terrific proposal that would have given the parties another year or so to negotiate an overall big deal, and well see if the Russians take it or not, O'Brien said.
After both Moscow and Washington withdrew from the 1987 Intermediate-range Nuclear Forces Treaty previous year, New START is the only nuclear arms control deal between the two countries still standing.
Mr. Putin has previously said he is ready to extend the current deal another five years, something he and Mr. Trump could do on their own.
Earlier, Russian Foreign Minister Sergey Lavrov voiced skepticism about reaching a deal on New START, noting that Russia can't accept the conditions put forward by the US for its extension.
The agreement can be extended for at most five years with the consent of the two countries.
The Trump administration's talk of a cap, or freeze, on all categories of warheads has puzzled some USA analysts, in part because the number of those weapons has held steady over the past decade or so.
After the last round of talks in Helsinki earlier this month, lead USA negotiator Marshall Billingslea, Trump's special envoy for arms control, said the meeting had yielded important progress.
US Secretary of State Mike Pompeo recently suggested that Washington intends to seek an early agreement on the treaty.
"It is clear that we have new weapons systems that the American side does not have yet", Putin said, adding that he is also willing to discuss this element. The US wants other weapons to be included in a new arms control deal.