The perpetrator - Martin Botzenhardt, a 48-year-old man from Munich - shot himself after crashing the silver-grey coloured van into the outside area of the restaurant, police said.
Broadcaster ZDF said police were searching his apartment and that he had contact with far-right extremists, but there was no evidence thus far that he was a far-right extremist himself.
The report said the man had wanted to revenge after the death of his accomplice Amri.
Following the incident, the interior minister of North Rhine-Westphalia said there was no indication that the attack was linked to Islamic terrorism.
Armin Laschet, the governor of North Rhine-Westphalia state, where Muenster is located, visited the crash scene on Sunday.
The man had in his possession two knives which had been especially sharpened for this objective, the report said.
German authorities do not believe there is a terrorist motive behind the attack, but it is believed the police searched the driver's apartment for explosives. "That is our current task", Bode said. "And the police arrived and everyone was sent out", he said.
A police spokeswoman said: "The danger is over".
"Unbelievable that something like this could happen in Muenster".
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Muenster University Hospital carried out several emergency operations on Saturday evening, according to Die Welt, but it is still unclear how many people sustained life-threatening injuries.
Witnesses said people ran away screaming from the city square after the crash.
Chancellor Angela Merkel said she was "deeply shaken" by the incident and "everything possible will be done to determine what was behind this act and to help the victims".
The newspaper says the main suspect allegedly knew a Tunisian who killed 12 when he drove a truck into a Christmas market in Berlin in December 2016.
In a statement, Merkel said "everything conceivable is being done to investigate the crime and to support the victims and their relatives". On Saturday evening, the White House issued a statement sending U.S. President Donald Trump's "thoughts and prayers" to the families of those killed.
The presidents of Russian Federation and France, Vladimir Putin and Emmanuel Macron, each sent their condolences.
Saturday's attack in Muenster came a year to the day of a truck attack in Stockholm in which a suspected Islamist militant sympathiser links killed five people. "France shares in Germany's suffering".
The half-marathon was being guarded by some 630 police officers, German news agency dpa reported.