Mexican Football Federation president Yon De Luisa said he was not considering resigning as the pressure builds following their failure to reach the knockout stages at the World Cup for the first time since 1978.
The North American side earned a 2-1 win over Saudi Arabia on Wednesday, but failed to qualify for the round of 16 on goal difference in Group C as Argentina and Poland went through.
De Luisa and the Mexican League's head Mikel Arriola, had been blamed for poor results even before the World Cup and the decisions to get rid of promotion and relegation in the Mexican league and allowing a large number of foreign players to play there.
"There's no need for a resignation, the (Mexican club) owners will make the decision, it's the end of a World Cup cycle," De Luisa told reporters.
"We were just one goal away from avoiding failure. Structural changes will be made, such as the number of foreigners (in the league), promotion and relegation."
Political pressure has grown, however, as Senator Bertha Caraveo demanded that De Luisa and Arriola be held accountable, calling for the executives to appear before the Senate's Youth and Sports Commission after the team's poor performance in Qatar.
"We will host the 2026 World Cup, and there will be immense logistical deployment and investment in its development, while the managers benefit while delivering poor results to the fans," Caraveo wrote on Twitter.
Mexico midfielder Orbelin Pineda did not hide his frustration after their elimination.
"We have long faces, we deserved something more, but that's football," AEK Athens player Pineda, 26, told ESPN.
"It is very important for players to go to Europe, because it stabilises their football (skills)."
"Hopefully, many Mexicans, like me, will fulfil their dream and be able to travel abroad. It is a very important growth for us and for Mexican football."