Putin also warned that a U.S. strike against Syria because of chemical weapons use could unleash new terrorist attacks. And he still maintained there is "every reason to believe" the weapons were used by rebels and not by Assad's military.
Obama, for his part, said simply that he was hoping for "a concrete result" from the talks.
The back-and-forth was a stark indication of the challenging work ahead as Kerry, Lavrov and their teams of chemical weapons experts plunge into talks aimed at finding agreement on how to dismantle the chemical weapons amid the confusion and danger of Syria's civil war.
Lavrov seemed to contradict Kerry's negative view of Syrian President Bashar Assad's offer to provide details on his country's chemical arsenal beginning 30 days after it signs an international convention banning such weapons.
The Russian said the initiative must proceed "in strict compliance with the rules that are established by the Organization for the Prohibition of Chemical Weapons." That suggests Russia does not agree with the U.S. that this is an exceptional case and that Syria should face tougher standards than other countries.
"We proceed from the fact that the solution to this problem will make unnecessary any strike on the Syrian Arab Republic, and I am convinced that our American colleagues, as President Obama stated, are firmly convinced that we should follow a peaceful way of resolution to the conflict in Syria." Lavrov said.
The distrust in U.S.-Russia relations was on display even in an off-hand parting exchange at the news conference.
Just before it ended, Kerry asked the Russian translator to repeat part of Lavrov's concluding remarks.
When it was clear that Kerry wasn't going to get an immediate re-translation, Lavrov apparently tried to assure him that he hadn't said anything controversial. "It was OK, John, don't worry," he said.
"You want me to take your word for it?" Kerry asked Lavrov. "It's a little early for that."
They were smiling at that point.
Shortly after making their opening statements, the two went into a private dinner. Talks were to resume Friday.
The meetings in Geneva got underway as Assad, in an interview with Russia's Rossiya-24 TV, said his government would start submitting data on its chemical weapons stockpile a month after signing the convention. He also said the Russian proposal for securing the weapons could work only if the U.S. halted threats of military action.
But Kerry, who met earlier Thursday with Lakhdar Brahimi, the U.N.-Arab League envoy for Syria, made clear the threat remains.
"President Obama has been clear that should diplomacy fail, force might be necessary to deter and degrade Assad's capacity to deliver these weapons," he said. "It won't get rid of them, but it could change his willingness to use them."
Even as diplomacy took center stage, word surfaced that the CIA has been delivering light machine guns and other small arms to Syrian rebels for several weeks, following Obama's statement in June that he would provide lethal aid to the rebels.
The U.S. is hoping that an acceptable agreement with the Russians can be part of a binding new U.N. Security Council resolution being negotiated that demands that Syria's chemical weapons be put under international control and dismantled and condemns the Aug. 21 attack that led to the current crisis. Russia has long opposed U.N. action on Syria, has vetoed three earlier resolutions and has not indicated it is willing to go along with one now.
As for arming the Syrian rebels, White House National Security Council spokeswoman Bernadette Meehan said the administration could not "detail every single type of support that we are providing to the opposition or discuss timelines for delivery, but it's important to note that both the political and the military opposition are and will be receiving this assistance."
Current and former U.S. intelligence officials said the CIA has arranged for the Syrian opposition to receive anti-tank weaponry such as rocket-propelled grenades through a third party, presumably one of the Gulf countries that have been arming the rebels. They spoke on condition of anonymity because they were not authorized to discuss the classified program publicly.
Loay al-Mikdad, a spokesman for the Free Syrian Army, told The Associated Press that his group expected to receive weapons in the near future.
"We are cooperating with the American administration and have been receiving some logistical and technical assistance and there are commitments by the administration to arm us but until now we have not received any weapons," al-Mikdad said by telephone.
The U.S. officials said the aid has been arriving for more than a month.
The U.S. team with Kerry in Geneva includes officials who worked on inspection and removal of unconventional weapons from Libya after 2003 and in Iraq after the first Gulf War. Officials with Kerry said the teams that eventually go into Syria would have to be an international mix.
The meetings are taking place in the same hotel where former Secretary of State Hillary Clinton in 2009 gave Lavrov a symbolic "reset button" as a goodwill gesture and a reminder of the Obama administration's efforts to improve U.S.-Russian relations.
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Associated Press writers Alan Fram and Kimberly Dozier in Washington, Vladimir Isachenkov in Moscow, Edith M. Lederer at the United Nations and Barbara Surk and Zeina Karam in Beirut contributed to this report.
