Denmark’s largest bank is investigating whether companies with ties to Russia used it to launder money, examining $150 billion in transactions that flowed through a tiny branch in Estonia, according to people familiar with the matter.

The $150 billion figure, covering a period between 2007 and 2015, has been presented to the bank’s board of directors and would equal to more than a year’s worth of the corporate profits for the entire country of Russia at the time. The flows would have stayed in the branch for only a short time...