QUITO, Ecuador—Vice President Joe Biden called Rafael Correa, the president of Ecuador, on Friday to ask that he reject a political-asylum request from Edward Snowden, who has admitted to leaking details of secret U.S. surveillance efforts.

The president of the small South American nation spoke of the conversation during his regular Saturday television broadcast, and told viewers that no decision would be made on Mr. Snowden's asylum request unless he was on Ecuadorean territory. If that happened, Mr. Correa said, he would...