The plan by Lon Snowden was revealed Wednesday after the father of the former National Security Agency (NSA) contractor gave an interview to Russian television and said he wanted to see his son to be safe.
"Today we agreed over the telephone (together with Edward Snowden) that I will put together an invitation for his father to come to Russia," lawyer Anatoly Kucherena told Russian news agencies.
"I hope that it won't take too long to issue a visa," he said.
Snowden, 30, is wanted on felony charges by the United States after leaking details of vast US surveillance programmes, but Russia has refused to extradite him.
More From This Section
Under mounting pressure over the leaks, Washington Wednesday declassified some details of its phone-tapping scheme.
Three secret documents, including a court order authorising the collection of US phone records, were released "in the public interest", according to Director of National Intelligence James Clapper. The order spells out how the government can use call data obtained from telecom giants such as Verizon.
He said he did not refuse the offer outright, however, and was considering the possibility of travelling to Russia and even attending the 2014 Winter Olympics in Sochi.
Speaking from a Washington studio, he greeted his son, after the presenter said he had been alerted to watch the show, and told him to stay safe.
"Edward, I hope you are watching this. Your family is well. We love you. We hope you are healthy, we hope you are well, I hope to see you soon, but most of all I want you to be safe. I want you to find a safe haven," Lon Snowden said.