The Qataris were arrested in connection with the assassination of former Chechen president Salim Khan Yandarbiyev in the Qatari capital Doha last month.
"The two Qatari citizens, Nassir Ibrahim Saad al-Madhihiki and Ibrahim Ahmad Nasir Ahmad, who represent Qatar's wrestling federation, are still detained by the Russian authorities," the foreign ministry said in a statement carried by the official QNA news agency.
"The Russian foreign ministry did not provide any details on the reasons for their arrest and their continued detention, although it promised clarification [last] Monday," it said.
"Russia did not authorise Qatar's consul in Moscow to visit the two detained Qataris despite repeated demands by the state of Qatar," the statement said, adding that "contacts are continuing with the Russian side."
The Belarus consul in Moscow, Viatchesav Streltsov, announced on Tuesday that one of the two had been released, adding that he was a citizen of Belarus and naming him as Alexander Dubrovski.
Streltsov also identified the second detained man as Ibad Akhmadov, who he said was arrested with a Qatari passport.
Qatar demanded on Sunday that Russia explain why two Qataris were arrested in Moscow after two Russian security agents were charged in Qatar with assassinating Yandarbiyev.
Qatari sources said the two men were officials with a judo team transiting through a Moscow airport en route to a tournament in Serbia when they were detained.
The former Chechen president
was killed in a car explosion
Moscow demanded on Friday that Qatar free its agents after denying they were involved in a blast that killed the 51-year-old and badly wounded his son outside a mosque.
Moscow's foreign ministry on Friday summoned Qatar's ambassador to hear protests against the arrests for the second time in two days.
Moscow's SVR foreign intelligence service has said it had nothing to do with the death of Yandarbiyev, who briefly headed Russia's war-torn separatist republic of Chechnya in the mid-1990s.