“Only the Syrian people define who’s going to be the president, when to come, and when to go. They never said a single word regarding this,” he said.
Assad’s fate is a key question in efforts to bring about a negotiated settlement to Syria’s five-year civil war.
But there has been little progress towards a resumption of talks that had been expected to take place this month.
And the prospects for a political transition beginning by August, as laid out in the plan, now appear slim.
Kerry arrived later on Thursday in Moscow, a close ally of Assad’s government that launched air strikes in support of regime forces last September.
He met with Putin at the Kremlin and both said before the meeting they hoped they could make progress on Syria. Kerry was also to meet Lavrov on Friday.
“I hope after today’s consultations you’ll be able to advise [US President Obama] of the progress made and possible headway for us to make,” Putin told Kerry, according to a pool reporter at the start of the talks.
For his part, Kerry told Putin: “Hopefully we’ll be able to make some genuine progress that is measurable and implementable and that can make a difference in the course of events in Syria.
The Washington Post reported on Thursday that Washington was to offer to cooperate with Russia in joint military action against the Islamic State of Iraq and the Levant (ISIL, also known as ISIS) and the Syrian branch of al-Qaeda, al-Nusra Front.
“The suggestion is that there’s going to be a plan on the table for the US and Russia to get together with air strikes. According to the leak, the detail suggests there will be active cooperation with flights and all attacks, and a ‘joint implementation group’ – as it has been described in the document,” said Al Jazeera’s Andrew Simmons, reporting from Moscow.
Sergey Karagonov, a former adviser to the Russian president, told Al Jazeera: “The problem is that he [Kerry] represents a lame duck. The general mood in the US is very negative towards Russia, towards cooperation.”
He added that while Lavrov and Kerry did not want to “exacerbate things together … it does not look like the problem will be solved easily”.
White House spokesman Josh Earnest said on Thursday that while there was “some speculation that an agreement may be reached”, it was “not clear that that will happen”.
“At present, the United States is not conducting or coordinating military operations with Russia,” he said.
Kerry’s visit to Moscow came amid fresh concerns over food aid in Syria. The UN on Wednesday said that along with other aid agencies, it has enough food in rebel-held eastern Aleppo to feed 145,000 people for one month, as pro-government forces continued to make progress on encircling and besieging the area, which has a population of as many as 300,000 people.
Kerry ‘extremely frustrated’
Kerry’s spokesman John Kirby told reporters his boss was “extremely frustrated” with the failure of peace efforts and “his patience was growing thin”.
But Kirby insisted the administration is not being naive, and that Thursday’s visit to Moscow, Kerry’s third this year, would “probe the sincerity” of Putin’s promises.
Syria’s conflict began in 2011 with the repression of anti-government demonstrations and has evolved into a complex multi-front war that has killed more than 280,000 people and forced millions from their homes.
The group has committed widespread atrocities in areas under its control and organised or inspired a wave of attacks across the Middle East and in Western cities.
Colvin ‘responsible’ for her own death
According to the Post, which cited sections of what it said was a draft agreement, US and Russian commanders would set up a joint command and control centre to direct intensified air strikes against the groups.
Such a deal is likely to face criticism that it amounts to a tacit acceptance of Putin’s efforts to shore up Assad’s regime.
In his interview with NBC, Assad also said a US reporter killed in alleged Syrian government bombardment in 2012 was responsible for her own death.
Marie Colvin, a 56-year-old war correspondent working for British newspaper The Sunday Times, died in the rebel-held Baba Amr district of Syria’s third city Homs.
“It’s a war and she came illegally to Syria. She worked with the terrorists, and because she came illegally, she’s been responsible of everything that befall on her,” Assad said, speaking in English.
Asked if she was responsible for her own death, Assad replied “of course”, though he denied that his forces had targeted her.
His comments came days after relatives of Colvin filed a case in a US court alleging Assad’s regime targeted her to stop her covering government atrocities.