Venezuela and Nicaragua had already offered Snowden refuge.
Bolivian President Evo Morales joined them, The Financial Times reported.
"I just want to say to the Europeans and Americans: we are going to give asylum if that American who is haunted by his countrymen asks us for it. We have no fear," Morales said.
Snowden has been stranded for several weeks in a transit area at Moscow's Sheremetyevo International Airport. He has made asylum requests to about 20 countries but most have either turned him down or told him he can only make the request once he is on their soil, RIA Novosti reported.
Nicaraguan President Daniel Ortega had previously extended a similar offer, saying he would grant Snowden asylum "if the circumstances permit," CNN reported Saturday.
Alexei Pushkov, head of the Duma, Russia's lower legislative body, applauded Venezuela's offer.
"The country has a sharp conflict with the United States," Pushkov said.
Snowden has been staying at the Moscow airport since departing Hong Kong June 23.
Birgitta Jonsdottir, a lawmaker in Iceland, announced "with great grief" that Snowden would not be able to get citizenship there because parliament had adjourned after refusing to consider a bill asking for his asylum request to be processed. She praised Snowden as a "hero" for revealing information about secret U.S. intelligence-gathering programs.
In offering Snowden asylum Friday, Venezuelan President Nicolas Maduro said he wanted "to protect this young man from the persecution unleashed by the world's most powerful empire," The New York Times reported.