Mexico, among the favourites and counting on massive support in the tournament in the United States, were crushed by Chile in their quarter-final in Santa Clara, California, last weekend.
FMF general secretary Guillermo Cantu told a news conference Osorio's departure was not a good option and the Colombian would continue trying to lead Mexico to the 2018 World Cup finals in Russia, the task for which he was appointed last October.
The defeat, called the most bitter and humiliating in Mexico's soccer history by media across the Rio Grande, ended a run of 22 matches unbeaten for the Mexico team.
Osorio's record in charge is nine wins and a draw in 11 matches.
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Chile will meet Argentina in the tournament final on Sunday.
(Reporting by Carlos Calvo; Writing by Rex Gowar; Editing by Ian Chadband)
