After their fifth loss in the Round of 16, the Mexicans have still only been as far as the quarter-finals on home soil - in 1970 and 1986 - and now have to decide if Aguirre is the man to stay around as long as Brazil 2014 amid rumours he may head back to Europe.
"We will speak with the Federation and inform them as soon as possible," was all the former Atletico Madrid coach would say after the 3-1 loss to the Argentinians at Soccer City Stadium in Johannesburg.
Aguirre himself says the job is long-haul by its nature. "We have to choose a coach for a four-year cycle and be patient as he moulds and matures these lads, while continuing with the structures we have, which are very good.
"They are doing things well at the Federation," insisted Aguirre, who took charge for the second time 13 months ago after Sven-Goran Eriksson's brief spell had produced poor results and almost cost the Mexicans qualification.
"I have been working for 13 months. Ideally, one would work for the Mexican Federation for 48 months. It is not easy to start all over again every four years," said Aguirre, who was in charge of them at the 2002 finals when they went out in the second round, suggesting he is ready to stay put.