Venezuela's offer of asylum could be NSA whistleblower Edward Snowden's "last chance" to avoid extradition to the US, an influential member of the Russian parliament suggested Sunday.

Alexei Pushkov, who chairs the Duma's foreign affairs committee and is seen to have the ear of the Kremlin, made the comment in a tweet, having previously stated his views that a move to Caracas would be the "best solution" for the fugitive.

Snowden is believed to be holed up in transit area of Moscow's Sheremetyevo airport, having arrived from Hong Kong two week's earlier but unable to progress further.

But on Friday and Saturday, three South America countries – Bolivia,Nicaragua and Venezuela – offered asylum to the man wanted in the US for leaking documents unveiling the extent of its domestic and international surveillance programme.

The US highest ranking military official acknowledged Sunday that Snowden's disclosures had already damaged US relations with some allies and affected "the importance of trust".

The chairman of the Joint Chiefs of Staff, General Martin Dempsey, told CNN that the US will "work our way back. But it has set us back temporarily."

Also Sunday, US House Intelligence Committee Chairman Mike Rogers said he "absolutely" thinks that one of the countries offering asylum will give Snowden the papers he needs to escape extradition.

He told CNN the US should look at trade agreements with the nations that are offering asylum "to send a very clear message that we won't put up with this kind of behavior".

The US has sent Venezuela an extradition request in advance of a potential move by Snowden to travel there.

In it, US authorities claims that the former National Security Agency contractor "unlawfully released classified information and documents to international media outlets" naming both the Guardian and the Washington Post.

Dated 3 July, the extradition request continues: "The United States seeks Snowden's provisional arrest should Snowden seek to travel to or transit through Venezuela. Snowden is a flight risk because of the substantial charges he is facing and his current and active attempts to remain a fugitive."

Venezuela president Nicolas Maduro has said that the request from the US has already been rejected.

On Friday both he and Nicaraguan counterpart Daniel Ortega offered asylum to the fugitive. It came as Latin American leaders condemned a decision to divert a plane carrying Bolivian president Evo Morales amid suspicion that Snowden might be on board.

Morales on Saturday also offered asylum to the young US whistleblower. But it is thought that Venezuela could be Snowden's best option.

There has been no official response from Moscow in regards to the asylum offers.

But Pushkov's comments suggest that some influential voices in Russia are anxious for a solution. The Russian parliamentarian has previously said that Snowden can not "live" at the Moscow airport.

In a tweet Sunday he went further: "Venezuela is waiting for an answer from Snowden. This, perhaps, is his last chance to receive political asylum."

But even if Snowden accepted the offer of asylum from Venezuela, problems getting to South America would continue to exist.

The only direct commercial flight from Moscow goes to Havana, Cuba, through European and US airspace.

Pushkov joked that if Snowden does not find shelter in Venezuela, "he will have to stay and marry Anna Chapman" – a reference to the redheaded Russian spy who was among 10 sleeper agents deported from the United States in 2010.

The 31-year-old Chapman proposed to Snowden, who just turned 30, on Twitter last week.