The African nation endured a turbulent campaign in Brazil as they lost all three of their matches in Group A.
Cameroon were beaten 1-0 by Mexico in their opening game and then conceded four goals to both Croatia and Brazil.
Their participation threatened to descend into farce even before their opening match against Mexico, as a row over bonuses saw the team arrive late in Brazil.
Further to the poor performances, Alex Song received a three-match ban for elbowing Croatia's Mario Mandzukic in the back, while Benoit Assou-Ekotto directed a headbutt at team-mate Benjamin Moukandjo during the same match.
The team arrived in Yaounde to a hostile reception on Wednesday and sports minister Adoum Garoua immediately called for an "urgent" assessment, a call which has now been echoed by the country's president.
A statement on Cameroonian radio said: "The President of the Republic, President (Paul) Biya, instructs the Prime Minister head of government Philemon Yang to open investigations on the causes of the unfortunate participation of the Indomitable Lions at the 2014 World Cup in Brazil.
"He should provide results within a month and make proposals for a deep and urgent reconstruction of Cameroon football."
