President Vladimir Putin's spokesman, Dmitry Peskov, said on July 2 that Snowden withdrew his application after he learned about the terms Moscow had set out.
On July 1, Putin said Snowden was welcome in Russia but said he must stop "inflicting damage on our American partners."
News media report that Snowden has sent asylum requests to 20 other countries, in addition to Russia.
India said that it had received a request but rejected it.
"Earlier today, our embassy in Moscow did receive a communication from Mr. Edward Snowden," Foreign Ministry spokesman Syed Akbaruddin told reporters in New Delhi. "That communication did contain a request for asylum. We have carefully examined the request. Following that careful examination, we have concluded that we see no reason to accede to that request."
Germany said it would consider the request it received.
"I can confirm that Germany has received at 8:11 a.m. local time in Moscow an asylum application by Mr. Snowden by fax," German Foreign Minister Guido Westerwelle said in Berlin. "We will review it thoroughly according to the legal frameworks. And I made sure that the application went promptly to the appropriate German authorities. They will review that application according to the law."
Snowden is wanted by Washington for leaking documents about U.S. surveillance programs.
He is reportedly staying at a Moscow airport hotel.
The European Union on July 2 reiterated its demand that Washington clear up the nature of its espionage program, which Snowden's leaks indicate targeted the EU and several of its member states.
"The [European] Commission expects clarity and transparency from partners and allies," European Commission President Jose Manuel Barroso told the European Parliament in Strasbourg, "and this is what we expect from our United States partners."
Meanwhile, Russian Foreign Minister Sergei Lavrov met with U.S. Secretary of State John Kerry on the sidelines of an ASEAN meeting in Brunei.
Lavrov said he did not discuss the Snowden case with Kerry since Putin had already said "everything" about it.