Halilhodzic guided Algeria beyond the group stage for the first time in their history in Brazil, but the round of 16 defeat against Germany on Monday looks set to be his last at the helm after his contract expired.
Algeria returned to a hero's welcome as thousands lined the streets of Algiers to greet Halilhodzic and his squad as they embarked on an open-top bus ride.
And Algeria president Bouteflika wants Halilhodzic, who has been linked with a return to Tukish club Trabzonspor, to build on his impressive work with the national team by staying at the helm.
He said in the presence of Algerian Football Federation (FAF) president Mohamed Raouraoua and Halilhodzic: "Vahid should stay with us. This is a great team we have."
Halilhodzic responded by saying: "It has not always been easy but we must keep on progressing. Thanks to our game, we have gained many fans in Brazil."
Christian Gourcuff has been touted as a potential successor to Halilhodzic.
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