where he had been staying for more than a week in legal limbo, a foreign ministry official told AFP.
"At 10:30 pm (1830 GMT) yesterday, British citizen Sarah Harrison turned up at the consulate department at Sheremetyevo airport and submitted a request from Snowden about granting him asylum," consulate officer Kim Shevchenko told AFP.
Sarah Harrison is an employee of anti-secrecy group WikiLeaks, who accompanied Snowden on his June 23 trip from Hong Kong, where the former NSA contractor leaked details of the US surveillance program after leaving his job in Hawaii.
Putin appeared to respond to Snowden's request on Monday by saying at a news conference that Snowden, who is wanted in the United States on charges of espionage, must stop leaking information damaging Washington if he wants to stay in Russia.
"Russia never hands over anybody anywhere and has no intention to do so," Putin said when grilled by reporters about the fate of the leaker believed to be holed up in the airport's transit zone since his arrival from Hong Kong on June 23.
"If he (Snowden) wants to remain here there is one condition - he should stop his work aimed at inflicting damage on our American partners no matter how strange this may sound coming from me," Putin said.
But Putin, whose comments came before the announcement by the consular officer that Snowden had applied for asylum, himself appeared to indicate that was an unlikely scenario.
"Because he feels like a rights activist and defender of human rights all indications are that he is not going to stop this work. So he has to choose a country of residence for himself and move there."
The Russian president reiterated Snowden was not a Moscow agent and was not working with Russian special services.
US President Barack Obama on Monday confirmed that there were high-level consultations between Moscow and Washington over Snowden's fate.
"We have gone through regular, law enforcement channels in enforcing the extradition request that we have made with respect to Mr Snowden," he said while on his African visit.
"Mr Snowden, we understand, has traveled there without a valid passport, without legal papers. We are hopeful that the Russian government makes decisions based on the normal procedures regarding international travel."
Putin had previously refused to immediately hand over Snowden to Washington due to the absence of an extradition treaty between the two countries.
The head of Russia's Security Council Nikolai Patrushev earlier Monday said that Putin and Obama had ordered the chiefs of their respective security agencies, the FSB and FBI, to find a way out of the diplomatic deadlock.
"Of course, (Putin and Obama) don't have a solution now that would work for both sides, so they have ordered the FSB director (Alexander) Bortnikov and FBI director Robert Mueller to keep in constant contact and find solutions," Patrushev told television channel Rossiya 24.
"I have to point out however that the task ahead of them is not easy, because they have to find a solution within the framework of international legal norms, and today one cannot say that such norms exist, and that there is a ready solution," Patrushev added.
Putin also hinted Monday Snowden might be swapped for another figure, saying that the only instance Moscow has given out people to other countries was "when we exchanged our intelligence officers for those people who were detained, arrested, and convicted in the Russian Federation."
Putin had last month reportedly sent a letter to Obama ahead of the G8 summit in Ireland, in which he listed a series of problems in ties with Washington.
The letter mentioned Russians Viktor Bout and Konstantin Yaroshenko, convicted and jailed in the United States, reported Kommersant broadsheet.
The two convicted Russian citizens could both be bargaining chips in the current negotiations between Washington and Moscow over Snowden, said Dmitry Trenin, the head of Moscow Carnegie Center think tank.
"If I was handling the negotiations, I would propose an exchange for Bout and Yaroshenko," he told AFP. "Another option is signing an agreement about mutual extradition, something the Americans don't want."
Snowden arrived in Moscow from Hong Kong on June 23 for a layover on his way to Latin America, possibly Ecuador, in a bid to escape extradition to the United States.
However, Ecuador President Rafael Correa said on Sunday that "the solution of Snowden's destination" was in the hands of Russian authorities.