A woman also under arrest in connection with the incident is also scheduled to be question, NRK reports.
“There will be heavy focus today on completing questioning of the two suspects,” senior police inspector Grete Lien Metlid said.
It was not possible to question the two suspects on Tuesday, Metlid said.
Several other individuals with connections to the two suspects will also be questioned, NRK writes.
The motivations for Tuesday’s incident were not immediately known but Oslo police said on Twitter they had “no information for now indicating that this is terror-related”.
“What was the purpose? What was the motive? We cannot answer this at such an early stage of the investigation,” Metlid said at a press conference.
“We are holding back for now on what happened and what may be behind it,” the senior officer also said.
“We have had several investigations and actions during the evening and night but I cannot go into what was seized,” she added.
Police earlier confirmed they had also arrested a woman linked to the male suspect accused of attempted murder.
Prior to the rampage, the pair — who both have criminal records — had been involved in another accident, during which their vehicle overturned.
When emergency services arrived at the scene, the man stole the ambulance while the woman fled.
“According to testimonies and observations, the suspect deliberately drove on the sidewalk to randomly hit victims,” Johan Fredriksen, deputy commissioner of the Oslo Police, said.
The incident took place in a residential neighbourhood in northern Oslo. Students at a nearby school and daycare centre were ordered to remain indoors for hours.
During his rampage, the man hit a woman with a pram carrying two seven-month-old twins, and an elderly couple had to throw themselves out of the way.
The woman and the babies had to be taken to a hospital but escaped without any severe injuries.
“Fortunately we can now confirm that the two twins… are considered to have minor injuries,” the Oslo University Hospital said in a Twitter post.
The man also fired a gun from behind the wheel of the ambulance at the officers dispatched to stop him.
After the ambulance crashed into a building, police arrested the 32-year-old Norwegian man who was already known to police.
Officers later also detained the woman, a 25-year-old Norwegian who appeared to be “under the influence” at the time of the first accident.
Investigators said they had found two firearms, one shotgun and an Uzi submachine gun, and suspected narcotics at the scene where the ambulance was stolen.
Police confirmed they were investigating reports by local media about the suspects links with far-right organisations.
The nature of the pair's relationship was not known.
In their attempts to stop the driver, police opened fire several times and the ambulance then crashed into a gate outside a residential building.
Television images showed a heavily damaged Mercedes ambulance with what appeared to be several bullet holes in the front left door.
A large number of police officers, some of them heavily armed, were deployed at the scene, and a helicopter hovered overhead for a while.
Norway has experienced several terrorist attacks by right-wing extremists, including the July 2011 twin attacks by Anders Behring Breivik that left 77 people dead.
However, there have been no attacks carried out by Islamist extremists on Norwegian soil.