US Secretary of State Antony Blinken speaks at a news conference, Wednesday, July 27, 2022 at the State Department in Washington. AP
The ongoing talks are one of the few areas where Moscow and Washington maintain dialogue as ties unravel over Russia's intervention in Ukraine and the onset of debilitating Western sanctions.
Russia and the United States have already conducted one prisoner swap since Moscow's Ukraine offensive. In April, Washington exchanged former US Marine Trevor Reed for convicted drug smuggler Konstantin Yaroshenko.
"A concrete result has not yet been achieved," Russian foreign ministry spokeswoman Maria Zakharova said on Thursday, referring to the ongoing talks.
Moscow hoped "the interests of both parties" would be taken into account, Zakharova said.
US Secretary of State Antony Blinken announced on Wednesday that Washington had made a "substantial proposal" to Moscow to free basketball star Brittney Griner and former US Marine Paul Whelan.
He declined to confirm reports that the United States was offering to trade them for convicted Russian arms trafficker Victor Bout.
"I have nothing more to add since there are no finalised agreements," Kremlin spokesman Dmitry Peskov told reporters.
Blinken said he would speak to his Russian counterpart Sergei Lavrov "in the coming days" -- in what would be their first telephone call since Moscow sent troops to Ukraine on February 24.
Bout, 55, is the highest-profile Russian prisoner in the United States.
He is serving a 25-year jail term and inspired the 2005 arms smuggling movie "Lord of War" starring Nicholas Cage.
He was dubbed the "Merchant of Death" by former British minister Peter Hain for supplying weapons to war-torn Angola and Liberia.
"Victor knows nothing about negotiations between Russia and the United States about his exchange," his wife Alla told state news agency RIA Novosti late on Wednesday.
"Of course, we assume that such negotiations might take place, but we don't talk about it because neither he nor I have any information."
Bout was accused over several decades of selling weapons to despots in some of the world's bloodiest conflicts.
The former Soviet air force officer and polyglot was convicted in 2011 of conspiring to sell arms to Colombia's FARC rebels after being arrested in Thailand in a US sting operation.
Whelan, a security official at an auto parts company, was arrested in Moscow and sentenced in 2020 to 16 years in prison for espionage, which he denies.
Griner, a WNBA champion and two-time Olympic basketball gold medallist, was detained in Russia just days before Moscow launched its offensive in Ukraine.
The 31-year-old athlete has pleaded guilty to drug charges for possessing vape cartridges with cannabis oil.
She is now on trial and faces up to 10 years in prison.
Griner's defence team said it learnt about the US exchange offer from news reports.
"The defence team is not participating in swap discussions," lawyer Maria Blagovolina said.
From a legal point of view, an exchange would be possible after the court reaches a verdict, she said, voicing hope that the US athlete could return home "soon".