A police statement said detectives would seek to trace Gaidar's movements before he fell violently ill during a conference in Dublin on November 24.
Gaidar's spokesman, Valery Natarov, says his boss is feeling better.
"There is no danger for his life, as the doctors say, and the health condition of [Gaidar] is improving," Natarov said.
Doctors in Moscow, where Gaidar is now recovering, said on November 30 that they saw no natural cause for his ailment.
His daughter says her father could be released from hospital as soon as next week.
The Kremlin's Fallen Foes
Mourners in Moscow mark the 40th day after the killing of investigative journalist Anna Politkovsksya on November 15 (TASS)
DANGEROUS DISSENT. A surprising number of vocal critics of the policies of Russian President Vladimir Putin have been killed or have died mysteriously in recent years. Although the Russian government has denied any involvement in any of the cases, some Kremlin watchers have begun speaking of a clandestine campaign to eliminate dissent.
April 17, 2003: Sergei Yushenkov, veteran liberal politician, Duma member, and leader of a staunchly anti-Kremlin party, is shot dead in Moscow.
July 3, 2003: Yury Shchekochikhin , liberal lawmaker and investigative journalist, dies of a mysterious allergic reaction. Many believe it was a case of deliberate poisoning, but the incident was never investigated as a murder.
February 13, 2004: Former acting Chechen President Zelimkhan Yandarbiyev is killed in an explosion in Doha, Qatar. Two Russian security-service agents are later convicted of carrying out the killing.
September 2, 2004: Investigative journalist Anna Politkovskaya falls ill mysteriously on a plane bound for the North Caucasus. Politkovskaya was heading to Beslan, North Ossetia, in a bid to negotiate the release of schoolchildren being held hostage there by Chechen militants.
December 2004: Former Ukrainian Prime Minister Viktor Yushchenko , running for president as a pro-Western candidate against Prime Minister Viktor Yanukovych, is poisoned. He recovers and goes on to win the presidency, although his poisoning remains a mystery.
October 7, 2006: Investigative journalist and vocal critic of Russian policies in the North Caucasus Anna Politkovskaya is gunned down in Moscow.
November 23, 2006: Former Federal Security Service agent Aleksandr Litvinenko, a vocal critic of Russia's secret services, dies of a mysterious poisoning in London.