President Barack Obama faced calls Sunday to pursue a more hawkish line on Russia, with an influential Republican foreign policy voice suggesting the US leader lacked sufficient insight over Vladimir Putin's intentions.

Arizona senator and former White House candidate John McCain suggested that comments made by Obama following the cancellation of a meeting with the Russian president did not go far enough to address a series of grievances Washington has with Moscow, including the handling of NSA whistleblower Edward Snowden.

Obama spoke on Friday of worsening US-Russia relations but said that he did not have a "bad personal relationship" with Putin, despite the tension suggested by his body language – "that kinda slouch, like a bored kid at the back of the classroom" – when the pair meet.

Speaking on Fox News Sunday, McCain said: "The president comparing him to a kid in the back of the classroom, I think, is very indicative of the president's lack of appreciation of who Vladimir Putin is."

"He's an old KGB colonel that has no illusions about our relationship, does not care about a relationship with the United States, continues to oppress his people, continues to oppress the media and continues to act in an autocratic and unhelpful fashion."

McCain said that cancelling the meeting – initially slated for September, but cancelled last week after Russia granted asylum to Snowden – was merely "symbolic".

He suggested that the US needed to do more, such as expand the 2012 Magnitsky Act, which seeks to punish Russians implicated in human rights abuses, as well as encourage Georgia's Nato aspirations and expand missile defence systems in Europe.

"We also need very badly to understand that Mr Putin does not have United States-Russia relationships as a priority" the senator said, adding that he needed to be treated in a "realistic fashion."

McCain said that by granting asylum to Snowden, Putin had in effect, put his finger in Obama's eye.

Snowden's disclosures of highly classified NSA programmes that revealed a vast US operations to access and monitor communications has caused tensions between the US and a number of foreign partners. Currently residing in Russia, he has since been charged in absentia of violations under the Espionage Act.

On Sunday, Snowden's father, Lon, said he had secured documents to visit his son in Russia and that he plans to talk to him about how he could fight the espionage charges against him.

He and Bruce Fein, the family's attorney, said that had not spoken directly with the former NSA contractor since he fled the United States. He added that he wants his son to return to America at some point.

But Lon Snowden, of Allentown, Pennsylvania, thinks it is unlikely that he could have a fair trial in the US, given what he called "absolutely irresponsible" descriptions of his son's actions from members of the Obama administration and top lawmakers from both parties.

"They have poisoned the well, so to speak, in terms of a potential jury pool," he said, speaking on ABC's This Week.

Fein said the family is willing to discuss conditions under which Snowden might return to the US and perhaps face criminal proceedings. They will provide him, they said, with suggestions of criminal defence attorneys with expertise under the Espionage Act, under which there have only been 10 prosecutions in as many years.

Fein said the planned trip to Russia would happen "very soon".

Snowden said that the political storm over his son's actions would fade and added "the American people are absolutely unhappy with what they've learned and more is forthcoming."

"Where my son chooses to live the rest of his life is going to be his decision. But I would like at some point in time for him to be able to come back to the US," Lon Snowden said.