The Sunday arrests followed the detention of two Russian security agents charged with assassinating a prominent Chechen separatist in the Gulf emirate.
Qatar's press dropped all diplomatic niceties used in the affair to date and claimed Moscow intended to use the pair as pawns to negotiate a tit-for-tat release of its own security men.
One foreign ministry official told the Qatar News Agency it was "in continual contact with the Russian foreign ministry demanding clarifications and an explanation.
"The Russian foreign ministry has replied that it will furnish an official response on Monday," the official added.
Qatari sources said the two men were officials with a judo team travelling through a Moscow airport en route to a tournament in Serbia when they were detained.
The Asharq newspaper ran an editorial headlined "Russian-style piracy" and charged Moscow with "jungle diplomacy".
The leading daily accused Moscow of seeking to use the men "to barter" for two Russian security service agents charged with assassinating Salim Khan Yandarbiyev in Doha on 13 February.
With the government owning 75% of the newspaper, the editorial read: "The Russians have not settled for creating a scandal over Yandarbiyev and have gone as far as piracy to achieve very stupid objectives.
"The Russians have not settled for creating a scandal over Yandarbiyev and have gone as far as piracy to achieve very stupid objectives"
"We are going to have a hard time with Russia which has thrown aside all international law which rules relations between countries."
But Moscow demanded last Friday that Qatar should free their two agents after denying involvement in the blast that killed the 51-year-old separatist and badly wounded his 13-year-old son outside a mosque.
A third Russian security agent was arrested and subsequently freed.
"Our main objective today is to obtain the freedom of our embassy staff and their return home," Deputy Foreign Minister Alexander Saltanov said.
"It's in the interests not only of the two men, but of Qatar and Russia and their bilateral relations."
Saltanov said the two secret agents had been posted to the Russian embassy in Qatar to help in the "fight against terrorism", in particular "terror financing".
The Russian foreign ministry on Friday summoned Qatar's ambassador to Moscow to hear Russia's protests about the arrests for the second time in two days.