Jose Manuel 'Chepo' de la Torre was dismissed on Saturday, one day after a 2-1 home defeat by Honduras in World Cup CONCACAF qualifying left the team staring elimination in the face.
Replacement Luis Fernando Tena, his assistant who led the country's Under-23 team to Olympic gold last year, then stepped down after Mexico lost their next qualifier 2-0 to the United States in Columbus on Tuesday.
"It's the most important challenge of my career ... hopefully the hot potato turns into gold," said Vucetich who is known as 'King Midas' after steering a string of Mexican clubs to league titles.
"It's a complicated situation I know but I also know it's an objective we can reach. There's confidence in the Mexican players," the 58-year-old told reporters.
Mexico are fifth in the group, level on points with fourth-placed Panama and with only a slim chance of making the top three automatic qualifying spots.
The United States and Costa Rica have already booked their tickets for the 2014 finals in Brazil.
Vucetich is Monterrey's most successful coach, having led the team from northern Mexico to the last three CONCACAF Champions League titles.
He also helped the club to two of their four domestic league crowns and won first division titles with Neza, Leon twice, Estudiantes Tecos, UANL Tigres, Cruz Azul and Pachuca.
Next up for Vucetich's Mexico will be a home game against Panama on October 11. Four days later they travel to Costa Rica for their last qualifier.
The team that finishes fourth in the group will take on Oceania winners New Zealand in a playoff for a place in Brazil.
Mexico, traditionally the strongest team in the CONCACAF region that covers North and Central America and the Caribbean, have only missed the World Cup finals once since 1982 and that was due to a ban in 1990 for fielding over-age players in a youth tournament.
Their campaign has been dismal and they have failed to make home advantage pay at the Azteca in Mexico City where the defeat by Honduras was only their second in that stadium in a qualifying tie.
Vucetich's team have eight points from eight matches, behind Panama on goals scored and three adrift of third-placed Honduras.
He was also in contention for the Mexico job in 2010 but dropped out for personal reasons.
Vucetich said he would name his squad on Sept. 23 for the October qualifiers and would choose only players from the domestic league.
"The mental aspect is the principal problem we face. We must know how to pick the players (whose minds) are not blocked in this regard," he added.
Mexico have called up half a dozen European-based players for their recent internationals including forwards Javier Hernandez and Giovani Dos Santos and midfielder Andres Guardado.
- Sports & Recreation