"I think Aguirre is going to have to show a keen eye to coordinate the team and pick the best players. Not getting past the first round would be a low blow for Mexican football," the gruff, mustachioed Argentine said.
Mexico, who have reached the second round at the last four World Cups, face a difficult task in Group A, playing hosts South Africa in the opening match on June 11, then 2006 runners-up France and Uruguay.
Aguirre earlier this week told the Spanish Cadena Ser he would quit the national team after the World Cup and return to coach in Europe because of growing insecurity in Mexico caused by drugs crime.
"There's no doubt that in the national team you've got to be careful what you say," La Volpe, a member of Argentina's 1978 World Cup-winning squad, was quoted on Friday as telling ESPN radio.
"I don't see Aguirre's thing as a surprise, I think his mind is set on carrying on coaching in Europe, he says so now and that's entirely his right."
La Volpe, who steered Mexico into the second round of the 2006 World Cup in Germany where they lost to his home country 2-1 after extra time, said he was prepared to take charge of the team again.
"If tomorrow I'm asked 'Would you coach the Mexico team?' of course I'd be right there because I believe I have the capacity," said La Volpe, who was in charge from 2002-06.
"I was the only one who lasted (four years), we went to an Olympics, the Confederations Cup, eight or nine players were sold to Europe after my tenure."
The 58-year-old La Volpe, recognised as an astute tactician, faced strong opposition as a foreigner in charge of the team from Mexican great Hugo Sanchez, who took over from him with mixed results.
Former England coach Sven-Goran Eriksson fared poorly in 2008-09 and Aguirre was recruited to steer Mexico through the latter stages of the qualifiers and into their fifth successive finals.
