The row comes down to UEFA's rule that no individual or company can have decisive influence over more than one club competing in a European tournament.
John Textor, the American businessman behind Eagle Football Group, owns significant shares in both Crystal Palace and Lyon.
UEFA ruled that Palace failed to show proof of ownership restructuring by the March 1st deadline.
As a result, Lyon remain in the Europa League, and Palace are set for the Conference League with Nottingham Forest taking their spot in Europe's second-tier competition.
Palace are seeking to have UEFA's decision overturned and to reclaim their Europa League spot, which would come at the expense of either Lyon or Forest.
The CAS confirmed Palace were seeking to take either Forest's or Lyon's place in the Europa League.
The club maintain that Textor has never had decisive control at Selhurst Park.
Palace chairman Steve Parish previously told The Rest is Football podcast: "We know, unequivocally, that John (Textor) did not have decisive influence over the club.
"We know we proved that beyond all reasonable doubt because it is a fact." UEFA, however, rejected Palace's defence.
A final verdict is expected on or before August 11th, ahead of the Europa League group stage, which starts on September 24th.
If the appeal fails, it is likely that Nottingham Forest - who finished seventh in the Premier League - will take Palace's place in the Europa League.
The case has raised fresh questions about UEFA's consistency in applying multi-club ownership rules.
Nottingham Forest's owner Evangelos Marinakis satisfied UEFA's criteria last year by putting his interests in a blind trust, while Palace's restructuring was deemed insufficient.
Textor has agreed to sell his stake in Palace to New York Jets owner Woody Johnson, but the move came too late to satisfy UEFA.
The outcome of the appeal could have significant consequences not only for Crystal Palace but also for the enforcement of multi-club ownership rules across European football, potentially affecting the owners of clubs including Manchester City and Manchester United.