Unlike Europe, in South America international cups (Libertadores and Sudamericana) are not played at the same time, so teams can qualify to both competitions without a problem.
Copa Libertadores runs from January/February to June while Copa Sudamericana starts in August and ends in December. Because of this year's tight schedule, Copa Libertadores will play semifinals and finals in July, after the World Cup but all teams from Argentina were already eliminated.
How do Argentinean teams qualify ?
For Copa Libertadores, 5 Argentinean teams qualify : 2 champions + 3 other teams.
For the 2011 edition, both 2010 champions qualify.
The other 3 spots are decided adding this 2 tournaments (Clausura 2010 and Apertura 2010), and the top 3 teams (not included the 2 champions) qualify.
Argentinos Juniors, recent winner of Clausura 2010 is the only team that has already qualified, and when Apertura ends in December, 4 other teams will join them.
For Copa Sudamericana the season table (Apertura 2009 + Clausura 2010) is used, and the top 6 teams qualify.
For the 2010 Copa Sudamericana that will start in early August, the qualified teams are Banfield, Argentinos Juniors, Estudiantes, Newell's, Independiente and Velez (qualified in that order by the 2009/2010 season table).
In the 1st round of the competition, Argentinean teams play between themselves : Banfield vs Velez, Argentinos Juniors vs Independiente and Estudiantes vs Newell's.
If you have any questions, please leave a comment or send me an email.