Kerry, speaking after talks with Lavrov and NATO colleagues in Brussels, also backed away from earlier comments suggesting he was calling for increased NATO contingency planning on Syria.
Kerry said he and Lavrov had discussed ways to revive a peace plan agreed in Geneva last June that called for a transitional government.
He said that while there might be a difference of opinion between Russia and the US about when and how Syrian President Bashar al-Assad might leave office, "I don't think there's a difference of opinion that his leaving may either be inevitable or necessary to be able to have a solution."
Moscow has for months been calling for implementation of the Geneva Declaration agreed by world powers including Russia and the United States, but disagrees with Washington's assertion that it requires Assad to step down.
Lavrov said last week that pressing for Assad's removal would increase the threats posed by militant Islamist groups such as the rebel al-Nusra Front, which formally pledged allegiance this month to al Qaeda leader Ayman al-Zawahri.
Lavrov, Kerry to talk Syria at Russia-NATO Council
The Russian partners are showing more understanding that the stake on continuing the war in Syria could lead to the growth of radical forces in the country, Russian Foreign Minister Sergei Lavrov said, Voice of Russia correspondent Olga Denisova reports.
"I feel more understanding of the urgency to switch from words and appeals to do something to real actions. I hope that we will soon see additional movements from those who doubted this previously," Lavrov said at a news conference on the results of talks with US Secretary of State John Kerry and a meeting of the NATO-Russia Council in Brussels.
In particular, "the understanding that if the stake on the war until the final victory supported by the minority of the international community will continue, then radicals will prevail," Lavrov said.
"The Syrian opposition currently has no united commandership, just separated groups. Militants of the cell the US put on list of terrorist organizations - these are the allies of those who put stakes on the war until the final victory - fight side by side with the so-called Free Syrian Army and groups, which obey to the so-called higher military council," the Russian foreign minister said.
"Our partners have a growing feeling that all this will end very badly, so we are currently searching for practical agreements, which will initiate the implementation the Geneva Communique," Lavrov said.
Lavrov, Kerry begin talks in Brussels
Russian Foreign Minister Sergei Lavrov is holding talks with US Secretary of State John Kerry on the sidelines of the Russia-NATO Council's ministerial meeting in Brussels on Tuesday.
It is the Russian and US top foreign policy officials' third meeting this year.
They last met in London two weeks ago. Kerry said he was glad to see Lavrov again.
The talks are expected to center on Russian-American cooperation and on international issues, including the situation in Syria.
Euro-Atlantic security agreement more important than NATO expansion - Lavrov
Russian Foreign Minister Sergei Lavrov thinks that NATO should not consider expanding, by accepting Georgia for instance, but agree with Moscow's idea to conclude a legally-binding agreement on security in the Euro-Atlantic region.
"It is necessary to think now not about expanding blocks with limited membership but about how to insure the all-European security. I'm deeply convinced that Georgia's security, security of any country of this region or any other Euro-Atlantic country will only benefit if dividing lines between military blocks are eliminated. And if all the countries without an exception will have equal security guarantees," Russian Foreign Minister Sergei Lavrov said at a news conference on the results of a meeting of the NATO-Russia Council, when asked about the prospects of Georgia joining NATO.
"And not like now, when your partners from NATO refuse the idea, our offer to conclude an agreement, in which repeatedly made political obligations not to enforce our own security at the expense of others' security would be obliging. They don't want that and that's it," Lavrov said.
"Then it turns out that they want to preserve the situation when there are kind of two types of guarantees as they understand: first-class guarantees for those who will be accepted in NATO and second-class guarantees for everyone else. Further expansion of NATO will only aggravate these risks," the Russian foreign minister said.
"I totally do not understand why Georgia needs this. I think that it is necessary to follow not the approach based on the principle 'who is against whom here and who can be against whom', this path is leading nowhere, this is an illusion, but the approach, which is aimed at positive things through positive things and not to reinforce guarantees of equal, same and indiscriminate basis through negative things," Lavrov said.