Russia appears in no rush
But Snowden could be a useful propaganda tool for Moscow, which accuses the United States of violating rights and freedoms it vocally urges other countries, including Russia, to protect.
Despite Putin and Lavrov's remarks, Moscow is clearly not in a hurry to dispatch Snowden from its territory. Ecuador, which has not in the past flinched from taking on Western powers, is similarly not rushing to banish the uncertainty plaguing US authorities.
The logical route for Snowden to take out of Moscow - and one for which he at one point had a reservation - would be an Aeroflot flight to Havana and a connecting flight to Ecuador.
But Ecuador's foreign minister indicated a decision on Snowden's asylum request could take two months.
"It took us two months to make a decision on Assange so do not expect us to make a decision sooner this time," Foreign Minister Richard Patino said in Kuala Lumpur, referring to the founder of anti-secrecy group Wikileaks, Julian Assange.
He added that Ecuador would consider giving Snowden protection before that if he went to Ecuador's embassy - but Russian officials say Snowden does not have a visa to enter Russia, and the United States has revoked his passport.
Snowden, who worked as a systems administrator at a US National Security Agency facility in Hawaii, was the source of disclosures about US government surveillance that included details about a program that collected emails, chat logs and other types of data from companies such as Google Inc, Facebook Inc, Microsoft Corp and Apple Inc .
He has divided opinion in the United States, where many have been outraged by the extent of government snooping.
A Reuters/Ipsos poll released on Wednesday showed that Americans were still more likely to view Edward Snowden as a "patriot" rather than a "traitor," but also that public support form him had fallen during the past week.
More than a quarter of respondents said Snowden should be prosecuted to the full extent of the law, up 3 percentage points from a week earlier. Just over one-third said he should not be prosecuted, down from a peak of more than 40 percent last week.
Russia's upper parliament house said it planned to investigate whether US intelligence agencies had violated the rights of Russians by collecting data from Internet companies.
A member of the Kremlin's advisory Human Rights Council, anti-corruption activist Kirill Kabanov, appealed to colleagues to ask Putin to invite Snowden to remain in Russia.
"We have shown that we are a weak country," state-run RIA quoted Kabanov as saying. "We could provide him with some kind of asylum. Surely we are not weaker than Ecuador."