SvD said the appointment had been approved by Volvo's two biggest owners, Renault and Industrivärden.
If the move is confirmed, Svanberg would replace Louis Schweitzer of France.
Svanberg is one of Sweden's top business executives, and has formerly served as the chief executive of telecom networks giant Ericsson and locks and security company Assa Abloy.
He has however been heavily criticised in his position at BP.
He had only been chairman of the British energy giant for a few months when a massive explosion on April 20, 2010 rocked the Gulf of Mexico Deepwater Horizon rig leased by BP.
Svanberg was lambasted for his low profile in the crucial weeks after the spill and Britain's Independent newspaper called him "the invisible man" while a Swedish daily referred to his "ostrich tactics."
Criticism also raged after a meeting with US President Barack Obama on the White House lawn when the Swede said BP "cared about the small people."
Should Svanberg move to Volvo, a Swedish switcheroo would be complete: Volvo's former chief executive Leif Johansson was recently appointed chairman of Ericsson.
The two men have long been considered Sweden's most successful business
Reflecting on the news, car industry expert Mikael Wickelgren at the University of Gothenburg said Svanberg's appointment makes a lot of sense.
”It wasn't really a surprise. Svanberg has been what you could call a management star in Sweden for a decade now and considering that Volvo had made the announcement of his predecessor's departure long ago it was quite a logical step to take for both sides,” he told The Local.
”What Svanberg can bring to the table is the benefit of his recent experience of working with a large international organisation. Even though it is too early to say if it is the right move or not, it can be seen a positive step for the company to have him at the helm.”
Despite inevitable protests, not least in the wake of Svanberg's handling of the BP oil spill crisis, Wickelgren doesn't believe his appointment will cause much of an upset.
”You could say it was a controversial appointment, but any candidate for such a position would have something that could be held against them” he added.
Volvo has so far refused to comment on the report.
"The company never comments on these kinds of questions," spokesman Mårten Wikforss told AFP.
When The Local contacted Volvo on Friday, the company confirmed they would not comment on shareholder issues.