Snowden is wanted for espionage, after leaking details of U.S. surveillance programs.
American authorities are demanding that Snowden be returned to the U.S. to face espionage charges.
In a weekly television address Saturday, Ecuador's president Rafael Correa said he told Vice President Biden he could not consider the request until Snowden arrived on Ecuadorean soil.
At the same time, Correa vowed he would consult with the U.S. if that happened.
"The moment that he arrives, if he arrives, the first thing is we'll ask the opinion of the United States, as we did in the Assange case with England," Correa said. "But the decision is ours to make."
The leftist leader avoided discussing Snowden's action but focused on addressing the U.S. spying secrets he exposed.
"The really grave thing is what Snowden has reported," Correa said. "He will have to assume his responsibilities, but the grave thing is his reporting of the biggest massive spy operation in the history of humanity, inside and outside the United States."
Snowden is believed to be holed up in a Moscow airport's transit zone and seeking safe passage to Ecuador.
The phone call between Correa and Biden took place Friday and was the first conversation to be publically disclosed since Snowden began seeking asylum.
A White House spokesman confirmed the call took place but would not disclose any further details about the conversation.
Snowden has with him four laptops with the U.S. government's most sensitive secrets.
ABC News and the Associated Press contributed to this report.