The black BMW left a trail of devastation when it tore through the traditional festive market in the centre of Magdeburg, southwest of Berlin in the formerly communist east Germany.
The local Volksstimme newspaper said reports from the scene indicated the attacker clearly tried "to hit as many people as possible".
Scholz and Interior Minister Nancy Faeser will on Saturday visit the market, where well-wishers had already left flowers of condolences.
Regional premier Reiner Haseloff said he would discuss the "necessary measures" to be taken with Scholz: "We now need to work through this and draw long-term consequences."
Police said the vehicle drove "at least 400 metres across the Christmas market" leaving behind a trail of destruction, debris and broken glass on the city's central town hall square.
The suspect, named by German media as Taleb A., was a psychiatric doctor who had lived in Germany since 2006 and held a permanent residence permit.
The Saudi government expressed "solidarity with the German people and the families of the victims", in a statement on social media platform X, and "affirmed its rejection of violence".
Series of attacks
Faeser, the interior minister, had recently called on people to be vigilant at Christmas markets, although she said that authorities had not received any specific threats.
Domestic security service the Office for the Protection of the Constitution had warned it considers Christmas markets to be an "ideologically suitable target for Islamist-motivated people".
Germany has in recent times seen a series of suspected Islamist knife attacks which have inflamed public opinion.
Three people were killed and eight wounded in a stabbing spree at a street festival in the western city of Solingen in August.
Police arrested a Syrian suspect over the attack that was claimed by IS.
In June, a policeman was killed in a knife attack in Mannheim, with an Afghan national held as the main suspect.
In the summer, the Scholz government imposed new border controls with European neighbours and pledged to step up deportations of rejected asylum-seekers.
Germany's conservative opposition leader Friedrich Merz, who is tipped to replace Scholz, has pledged in his campaign to show "zero tolerance" on crime and "stop illegal migration".