(CNN) — The Biden administration has had communications with Russia to try and secure the release of Brittney Griner and Paul Whelan "as recently as within past days" said a senior administration official, speaking to CNN on Griner's 32nd birthday, which she will be spending in a Russian jail.

The United States first put an offer for a prisoner swap on the table with Russia back in June -- the details of which CNN exclusively reported -- and "conversations have not been static since then" the official said.

Despite the "pretty persistent" pace of discussions between the US and Russia to secure the Americans' release, the official said that the Biden administration has yet to receive a serious counteroffer from the Russian side.

"We have worked hard to try to demonstrate the sorts of things that could well be the basis for resolving this and each time we have articulated that it's been met not with a serious counteroffer," the official said.

They said the Russians have countered with "something not in our control, not in our ability to deliver," but did not go into further specifics.

"They're not non-responsive. I would say that they continue to respond with something that they know not to be feasible or available," the official said of the Russian response.

The official said the US has used multiple channels for discussions with the Russians and conversations have taken place both in-person, by phone, and "through other forms." The US has dangled multiple ideas for "things that could be in play" to urge a serious response from the Russians.

As negotiations continue, Griner and Whelan both remain behind bars, with the WNBA player detained since February and the ex-Marine detained since December 2018.

Griner was sentenced to nine years in prison in August, and the US Embassy has not had consular access to her since then, State Department spokesperson Ned Price said last week. However, US consular officers spoke by phone "briefly" with Griner and Whelan, principal deputy spokesperson Vedant Patel said Tuesday.

Griner released a statement on her birthday Tuesday thanking "everyone for fighting so hard to get (her) home."

"All the support and love are definitely helping me," she said in the statement, which was shared by Maria Blagovolina, a partner at Rybalkin, Gortsunyan, Dyakin and Partners law firm.

The US senior administration said said that "every day is too long" for Griner to remain wrongfully detained.

"I wish you weren't spending this birthday in Russian detention. I wish you weren't spending the past weeks and months there," the official said in a message to Griner.

"As far as we're concerned, each day is too long and we will keep working this until we resolve it and get her home. Regrettably, the other side gets a vote in this. They're the ones who created this horrific situation. They're the ones we regrettably need to deal with to resolve it."

Next week Griner will appeal her 9-year prison sentence. It is unclear if the passage of that court date will bolster ongoing efforts to get her home.

"To the extent that that different phases of that decidedly imperfect system pass and open the possibility on the other side of real negotiations, we would welcome that. But the most candid answer is, we don't know," the official said.

President Joe Biden said last week that he would consider meeting with Russian President Vladimir Putin at the G-20 in November if he wanted to discuss Griner. When asked if there has been discussion between the US and Russia about that possible meeting on Griner the official said they would let Biden's remarks "speak for themselves."

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