Over the weekend, Correa said Vice President Joe Biden asked Ecuador to reject Snowden's request.
Obama said Monday that Snowden had traveled to Russia without a valid passport or legal papers and that he hoped that Moscow would handle the case as it would any other travel-related matter.
Here's a rundown of what countries are saying about Snowden's asylum requests:
• Austria: His request is not valid, because applicants must be in Austria and submit their applications in person.
• Bolivia: Despite WikiLeaks' statement, Morales, the Bolivian president, said Bolivia has not received a request for asylum on Snowden's behalf, Itar-Tass reported.
• Brazil: The country won't grant asylum or even respond to the request, Reuters reported Tuesday, citing a Foreign Ministry spokesman.
• China: The Ministry of Foreign Affairs said Tuesday it had no information to offer on Snowden's case.
• Cuba: No response yet.
• Ecuador: The South American country has said that it's considering Snowden's request but that he needs to reach its territory.
• Finland: "We don't have an official application from Snowden," Jorma Vuorio, director general for the migration department of the Interior Ministry, told CNN. "He just left a letter requesting asylum at the Finnish Embassy in Moscow. According to Finnish legislation, it is not possible to leave an asylum application outside the Finnish territory."
• France: Hollande, the French president, said he has "not received any particular demands from Mr. Snowden," according to French television reports.
• Germany: Asylum requests must be made from within the country, but an Interior Ministry spokesman also said "the German right of residence principally entails the possibility of acceptance from abroad, if this seems necessary for international legal or urgent humanitarian reasons, or for the ensuring of political interests of the federal republic of Germany. This needs to be examined thoroughly in the case of Mr. Snowden."
• Iceland: Officials have said asylum seekers need to be in Icelandic territory in order to apply.
• India: Ministry of External Affairs spokesman Syed Akbaruddin said Tuesday that India will deny the request.
• Ireland: The Irish Justice Ministry declined to confirm whether it had received a request on Snowden's behalf, but said Irish law requires that applicants reach the country's borders before a request can be considered.
• Italy: The Italian Foreign Ministry said it received an asylum request by fax. The country accepts requests made in Italian territory or at the border, the ministry said.
• The Netherlands: Snowden would need to be in Holland to formally request shelter, a spokesman for the Dutch state secretary said Tuesday.
• Nicaragua: No response yet.
• Norway: The country confirms the receipt of his request, but usually, asylum seekers must be in the country already to be allowed to apply. The Ministry of Justice is considering whether or not to process his application.
• Poland: Snowden's request for asylum doesn't fulfill requirements, the Ministry of Foreign Affairs said Tuesday.
• Russia: Snowden has withdrawn his request for permission to stay in Russia, Kremlin spokesman Dmitry Peskov said Tuesday, according to the news agency RIA Novosti.
• Spain: The country is not reviewing Snowden's request because it must be made from within or on its borders, a Foreign Ministry spokeswoman said.
• Switzerland: Applicants for political asylum have to be on Swiss soil, a spokeswoman for the Swiss federal office for migration said Tuesday. Swiss authorities aren't aware of any application by Snowden, the spokeswoman said.
• Venezuela: "We haven't been asked for help, but when he asks, he will have an answer," Maduro, the Venezuelan president, said Tuesday.