CONMEBOL issued a statement on Sunday (AEDT) confirming it will speak with FIFA and national associations to agree on new dates for the matches.
The decision was taken amid concerns around the logistics of players joining up with their national teams while travel restrictions caused by the coronavirus pandemic are in place.
CONMEBOL said it was "impossible" to guarantee all South American players could link up with respective squads in a timely manner.
The next two rounds of matches were scheduled to be held on 26. 27 and 31 March (AEDT).
The confederation will now study the options available in order to reschedule the matches.
FIFA has given clubs dispensation to prevent players who may be affected by coronavirus regulations from joining up with their countries during the upcoming international window.
Manchester City boss Pep Guardiola joined Liverpool counterpart Jurgen Klopp in threatening to stop his players from travelling for internationals if they would be required to quarantine on their return.
Under current coronavirus guidelines, players arriving in Great Britain from "red-list" countries – including Brazil and Argentina – are subject to 10 days' hotel confinement.
Brazil had been due to play Argentina in Recife on 31 March (AEDT), in a meeting of the top two countries in South American qualifying for next year's FIFA World Cup in Qatar.