Despite the reservations, hundreds still took to the streets of Algiers on Monday night, dancing, singing and honking vehicle horns in celebration.
Residents of the Capital Algiers blared their vehicle horns and flew the national flag as they sang "peacefully, we have overthrown the puppet!".
Secrecy surrounds Algeria's leadership and Bouteflika himself - it has never been clear whether full power lay in his hands, or whether army generals who molded the North African nation called the shots from offstage.
As Walid Laggoune, a professor of constitutional law at the University of Algiers, told public radio: "The constitution does not provide for any provision on the postponement of elections".
He says he plans to appoint a new government and a separate "national conference" tasked with rescheduling the election and drafting a new constitution.
A wily political survivor, Bouteflika fought in Algeria's independence war against French forces and has played a role in Algeria's major developments for the past half-century.
Bouteflika has named interior minister Noureddine Bedoui as the new prime minister to replace the unpopular Ahmed Ouyahia.
Algerian president returns home after 'routine medical checks' in Geneva
"It's a ploy to win time.in order to put in place another puppet like the (current) president", said a student who gave her name as Amel.
The United States said it backed the talks going on in Algeria and that it was "closely monitoring" reports elections had been postponed.
The protests have shattered years of political inertia and unsettled Algeria's opaque but powerful security establishment.
A number of protests have also been staged in Switzerland against the Algerian president since he arrived in the country.
The unprecedented citizens' revolt drew millions into the streets of cities across the country to demand that Bouteflika abandon his candidacy.
Mr Brahimi, who was the United Nations and Arab League Special Envoy to Syria until 14 May 2014, met the president on Monday and said it was necessary to "turn this crisis into a constructive process".
Bouteflika, who is 82, has barely been seen in public since a 2013 stroke. No sound accompanied the images.
The protests trickled down to middle-schoolers and high-schoolers, with several hundred marching in the centre of Algiers, also calling for Bouteflika to withdraw his bid for a fifth mandate.
Despite the government's efforts to contain the protests, some workers in the energy sector, which accounts for an estimated 30%of the country's GDP, joined the strike on Sunday, the first day of the working week in Algeria.
Cabrera Bello avoids water to set pace with 65 at Bay Hill
Baddeley is the leading Australian at Bay Hill, with 2017 Palmer victor Marc Leishman posting a 70 to sit at two under. British Open champion Francesco Molinari made one of two aces on the day and was among those at 69.