Abdelaali Hassani, who heads the moderate Islamist party the Movement of Society for Peace, said the day before he had "lost the battle but not the war" and denounced the results as a "fraud".
He was one of just two challengers to Tebboune in Saturday's election, the second being Youcef Aouchiche of the centre-left Socialist Forces Front (FFS), who is also expected to appeal.
The North African country's electoral authority, ANIE, announced on Sunday that Tebboune had won "94.65 percent of the vote", with Hassani on 3.17 percent and Aouchiche 2.16 percent.
Hassani earlier denounced what he called "false figures" on voter turnout and demanded that the authorities put an end to the "masquerade".
Tebboune, 78, had been widely expected to breeze through the election and was focused instead on securing a high turnout.
He was elected in December 2019 with 58 percent of the vote, despite a record abstention rate above 60 percent, amid the massive Hirak pro-democracy protests.
More than 24 million Algerians were registered to vote in this election, but ANIE did not say how many people turned out.
Instead, it announced a "provisional average turnout" rate of 48 percent, which many including Hassani and Aouchiche have disputed.
The Constitutional Court is set to announce the final results within 10 days of receiving the count from Algeria's 58 provinces.
On Monday it said it had yet to receive all of them.
