However, the postponement of the remainder of the second round of AFC qualifying for the World Cup until June because of the COVID-19 pandemic has led both teams to pull out.
Qatar, who announced their withdrawal on Tuesday, are competing in the qualifying process to secure a place in the 2023 Asian Cup. They are top of Group E and will seal their spot if they win it.
Australia are top of Group B and on course to qualify for the Asian Cup and progress to the third round of the World Cup qualification progress.
"The Socceroos' players and staff were greatly looking forward to the unique opportunity of playing in the Copa America and it is unfortunate that, due to the COVID-19 pandemic, we will be unable to attend in 2021," Australia coach Graham Arnold said.
"On behalf of the team I thank CONMEBOL for the invitation, which would have presented our team with a fantastic opportunity to test itself against some of the world's best."