The Interfax news agency, citing sources in Moscow, reported that Akayev arrived in the Russian capital from Kazakhstan on Friday night.
The Kyrgyz embassy in Moscow said it had no information on the ousted leader's whereabouts, while the Russian Foreign Ministry and the president's office refused to comment.
Russian President Vladimir Putin said on Friday that Moscow would not object if Akayev wanted to go to Russia.
Akayev's whereabouts had been a mystery, although a statement purportedly from him released on Friday said he was out of the country only temporarily, denied he had resigned, and denounced what he called the opposition's "unconstitutional coup d'etat".
The fall of the longtime leader's government came swiftly on Thursday after weeks of intensifying protests in the south, propelled by widespread anger over disputed parliamentary elections in late February and early March.
Akayev's wife, Mairam, confirmed that her husband fled the capital Bishkek as protesters began to storm the government building, Russia's Rossiyskaya Gazeta reported on Saturday.
"If you consider the fact that Putin invited him, it's perfectly feasible, but it creates problems for Moscow"
expert on Central Asia
The paper, which did not say where she was, quoted her as saying that Akayev left "by agreement with the European community", but she did not elaborate.
"We are safe, but the country is dying," she said.
Arkady Dubnov, a Central Asia expert for Russia's Vremya Novostei newspaper who enjoyed access to Kyrgyzstan's leadership, said he had no confirmation of the news but he told reporters he would not be surprised if it were true.
Storing up trouble
"If you consider the fact that Putin invited him, it's perfectly feasible, but it creates problems for Moscow," he said.
"Russia will have given refuge to a man who remains the constitutionally elected leader of Kyrgyzstan and has refused to step down. It is a humanitarian act, but it will make it difficult to recognise the new authorities there," Dubnov said.
He added that an additional headache for Russia was that the acting Kyrgyz Foreign Minister Roza Otunbayeva, signalled the new government might ask for Akayev's extradition when it became clear where he had fled to.
The vice-president of the Russian Academy of Sciences, Gennady Mesyats, was quoted as saying on Friday that the academy would be willing to give a job to Akayev, who is a Russian-trained physicist.