FIFA, global football's governing body, ruled for Ecuador in early September in a dispute over the nationality of Byron Castillo, who played for Ecuador in qualifying.
CAS is the last legal resort available to the two South American nations. Ecuador will kick off the World Cup in Doha on November 20 when they face hosts Qatar.
A court statement said "both appellants request that CAS issue its final award on or before 10 November 2022 at the latest." Castillo, a defender, played in eight of Ecuador's qualifiers, including both matches against Chile -- a win for Ecuador and a draw.
He did not appear in either of Ecuador's games against Peru, in which Peru picked up four points. The Chilean football federation (FFCH) said there are "countless proofs" that Castillo, who plays for Leon in Mexico, was born in Colombia in July 1995 and not in Playas, Ecuador in November 1998.