According to the Kommersant business daily on Friday, the two Qataris, arrested at Moscow's Sheremetyevo airport on 26 February, had been held by the Federal Security Service (FSB), but were recently put under the jurisdiction of Russia's foreign ministry ahead of the planned swap.
Foreign ministry press officials declined to comment on the report.
The Qataris were detained in Moscow shortly after Doha's arrest of two Russian intelligence agents, who are charged with assassinating former Chechen president Salim Khan Yandarbiyev, in the Qatari capital on 13 February.
Kommersant identified the two detained Qataris as Nasseihiki, an official with the Qatar Greco-Roman wrestling team and Ibad Akhmedov, an Azerbaijani who has Qatari citizenship and competes for the team.
Qatari officials however have identified the detainees as Nasser Ibrahim Saad al-Madhihiki and Ibrahim Ahmad Nasser Ahmad.
Russia's new Foreign Minister Sergei Lavrov on Wednesday reiterated Moscow's demands that Qatar free the two Russian agents.
Moscow has denied its agents had anything to do with the murder of Yandarbiyev, who died after his car blew up in Doha as he was returning home from Friday prayers.
Russia accused Yandarbiyev of helping finance separatist Chechen fighters who have been leading a guerrilla war in the predominantly Muslim republic since 1999.
Yandarbiyev briefly served as Chechen president at the end of the first, 1994-96 Chechen war.