The U.S. believes the massing of tens of thousands of Russian soldiers, ostensibly for military exercises, along the border is at once an attempt to intimidate Ukraine's new leaders after Russia's annexation of the strategic Crimean peninsula and to use a bargaining chip with the United States and the European Union, which have condemned Crimea's absorption into Russia and imposed sanctions on senior Russian officials.
Kerry noted that even if the troops remain on Russian soil and do not enter Ukraine, they create a negative atmosphere.
"The question is not one of right or legality," he said. "The question is one of strategic appropriateness and whether it's smart at this moment of time to have troops massed on the border."
U.S. officials said Kerry proposed a number of ideas on troop withdrawals from the border and that Lavrov, while making no promises, told him he would present the proposals to the Kremlin.
At a separate news conference at the Russian ambassador's house, Lavrov did not address the troop issue. Instead, he made the case for Moscow's idea of Ukraine as a federalized nation with its various regions enjoying major autonomy from the government in Kiev. Russia says it is particularly concerned about the treatment of ethnic Russians and Russian speakers who live in southern and eastern Ukraine.
Lavrov said that Ukraine can't function as a "unified state" and should be a loose federation of regions that are each allowed to choose their own economic, financial, social, linguistic and religious models.
He said every time Ukraine has elected a new president, the country has adopted a new constitution, proving that "the model of a unified state doesn't work."
Ukrainian officials are wary of decentralizing power, fearing that pro-Russia regions would hamper its Western aspirations and potentially split the country apart. However, they are exploring political reforms that could grant more authority to local governments.
The U.S. has been coy about their position on a federation. Washington has encouraged ongoing political and constitutional reform efforts that the government in Kiev is now working on but U.S. officials insist that any changes to Ukraine's governing structure must be acceptable to the Ukrainians.
Kerry said the federation idea had not been discussed in any serious way during his meeting with Lavrov "because it would have been inappropriate to do so without Ukrainian input."
"It is not up to us to make any decision or agreement regarding federalization," he said. "It is up to Ukrainians."
"We will not accept a path forward where the legitimate government of Ukraine is not at the table," Kerry said, adding that the bottom line is: "No decisions about Ukraine without Ukraine."
Lavrov denied that Moscow wants to "split Ukraine."
"Federation does not mean, as some in Kiev fear, an attempt to split Ukraine," he said. "To the contrary, federation ... answers the interests of all regions of Ukraine."
Lavrov said he and Kerry did agree to work with the Ukrainian government to improve rights for Russian-speaking Ukrainians and disarm "irregular forces and provocateurs."
Sunday's meeting was hastily arranged 48 hours after U.S. President Barack Obama and Russian President Vladimir Putin spoke by phone in a conversation in which Obama urged Putin to withdraw his troops from the border with Ukraine. Putin, who initiated the call, asserted that Ukraine's government is allowing extremists to intimidate ethnic Russians and Russian-speaking civilians with impunity something Ukraine insists is not happening.
That call did little to reassure U.S. officials that Russia is not planning to invade Ukraine after its annexation of Crimea that the West has condemned as illegal and a violation of Ukraine's sovereignty and territorial integrity. The United States and Europe have imposed sanctions on senior Russian officials in response, sparking reciprocal moves from Moscow.
In the interview with Russian television, Lavrov called the sanctions a "dead-end" strategy that would not achieve results and accused the West of hypocrisy. He said it was inconsistent for the west to refuse to recognize Crimea's annexation, which followed a referendum on joining Russia that was overwhelmingly approved, while at the same time accepting the new government in Kiev, which was formed after the pro-Moscow president fled the country.
The idea for Sunday's meeting was for Lavrov to present Russia's responses to a U.S. proposal to de-escalate the tensions that covers Ukrainian political and constitutional reforms as well as the disarmament of irregular forces, international monitors to protect minority rights and direct dialogue between Russia and Ukraine, according to U.S. officials, who say it has backing of Ukraine's government.
U.S. officials said Russia appeared to show some interest in the Ukrainian reform program, but Moscow's insistence on a federated state left it unclear if they would support it.
Kerry and Lavrov have met several times in person and have spoken by phone almost daily since the crisis began but have not yet been able to agree on a way forward. The pair met last week in The Hague, where Kerry presented Lavrov with the proposal, which was a response to ideas Lavrov gave him at a March 14 meeting in London.
Associated Press writer Angela Charlton contributed to this report.