The 82-year-old has come under mounting pressure to step down since his decision to seek a fifth term as president despite rarely being seen in public since a 2013 stroke.
Bouteflika said earlier this month he would pull out of the race and postponed April elections, in moves that angered Algerians who see it as a ploy to extend his two decades in power.
Tens of thousands have taken to the streets across the country demanding he quit since he announced he was standing for another term with the support of both his National Liberation Front and the National Rally for Democracy (RND).
In a statement signed by its leader, recently sacked prime minister Ahmed Ouyahia, the RND said on Wednesday that it "recommends the resignation of the president...with the aim of smoothing the period of transition."
Ouyahia is a longtime supporter of Bouteflika and served as his prime minister 3 times since 2003 before being sacrificed on March 11 in a vain bid to calm the intensifying street protests.
'Unfit to rule'
On Tuesday, March 26, armed forces chief of staff General Ahmed Gaid Salah, who was appointed by Bouteflika in 2004, said the president should either resign or be declared medically unfit to govern by parliament using its constitutional powers.
The general, who is also deputy defense minister, called for the application of Article 102 of the constitution that puts the onus on the president either to resign or be declared unfit to govern due to a "serious and durable illness."
The move would potentially clear the way for elections to be organized in the coming months unless the president recovers.
In its statement, the RND said it viewed "favorably" the proposal by the chief of staff of the army, which "seeks to preserve the country from blockage."
The party went on to pay "homage to Abdelaziz Bouteflika for all he has done for Algeria."
Bouteflika's stroke affected his mobility and speech, and in rare recent public appearances he was in a wheelchair.
The prospect of another 5 years of being ruled by a powerless fig leaf president sparked the eruption of mass protests on February 22 that have escalated with each successive week.
Key regime figures have since broken ranks with Bouteflika one by one in a bid to retain their hold on power.
For now there is nothing to force him to resign.
The Constitutional Council, whose head is a Bouteflika loyalist, is the only body allowed to launch a procedure to declare the president unfit for office.
But time is running out.
While he announced he was withdrawing his bid for a fifth term, Bouteflika also postponed the election that had been scheduled for April 18.
It is now due to be held after a "national conference" called to discuss reforms, followed by a referendum on a new constitution, a process that would effectively leave Bouteflika in power for an indefinite period.
That is something protesters strongly oppose.
"The RND is on the same side as the army" and they are both seeking to "save the system one month from the end of the mandate," said Mahrez Bouich, professor of political philosophy at the University of Bejaia.
But "the people do not want a simple change of clan" but rather "a radical break from the system," he said.
On the streets of central Algiers, where protests have been held every Friday to demand the departure of the Bouteflika regime, people remained cautious.
Efforts to remove the president were a "poisoned gift" because Article 102 would effectively keep the same system in place, said Ahcene Zenati, a 45-year-old economist.
"Nothing is concrete," said Yahia, a 64-year-old lab technician, and "the people could even ask for more" than Bouteflika's departure. – Rappler.com