Assange, 41, is seeking asylum in the South American nation to avoid his extradition to Sweden, where he is accused of sexual assault.
"The important thing is for Julian to be assured that Ecuador is considering with great responsibility" his request, the president told Assange's mother, Christine Assange, during their hour-long meeting.
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Correa said his country has a "great humanist tradition and respect for human rights."
He added that he respects Britain, Sweden, and the people of the United States, and that the matter bears consultation, but that, ultimately, "Ecuador does not negotiate over its sovereignty."
Meanwhile, Assange told AFP that based on "a chronological history of the Swedish investigations and based on the fact, there is absolutely no doubt" the charges against her son are part of a political persecution against him.
During her meeting with Correa, Assange expressed her "concerns and qualms about what could happen to Julian Assange if he is extradited to Sweden by the British government," Foreign Minister Ricardo Patino told reporters.
Assange sought refuge at Ecuador's embassy on June 19, fearing that he could be extradited to the United States from Sweden to stand trial for espionage, on account of the trove of leaked US diplomatic cables and military logs that were published on his whistleblower website.
The leak represented the biggest breach of US intelligence in history.
Patino has said that Ecuador will respond to Assange's request on August 12, after the London Olympics.
Correa has often been at odds with Washington and offered Assange asylum in 2010.
He vowed earlier this month that his government would not yield to pressure from Britain, Sweden or the United States in deciding whether to grant Assange asylum.
