The German authorities launched an inquiry into the allegations covering "bank officials in Germany, Switzerland and Luxembourg" earlier in the week, according to Spiegel.
Bank head Hans-Walter Peters, who has just taken over as pesident of the powerful German association of private bankers, is "in the (investigators') sights," Spiegel added.
A Berenberg spokesman told AFP by email that the bank was in contact with the prosecutor's office but added that no judicial procedure was underway against Peters.
The prosecutor's office was unavailable for comment.
Peters took over from outgoing Deutsche Bank co-CEO Juergen Fitschen at the head of the private bankers' association last week with the latter embroiled in a longstanding legal case involving the collapse in 2002 of the Kirch media empire.
Hamburg-based Berenberg is mentioned several times in the huge offshore Panama Papers data dump unveiled earlier this month by international media.
The bank said last week, "like many others," it had managed offshore accounts for clients but insisted it had done so "in line with the law."
Spiegel also returned to the Panama Papers theme in alleging that Finance Minister Wolfgang Schauble had "for years" not engaged with a reported whistleblower whom it said had offered to hand over evidence of wrongdoing regarding links between the Bundesdrueckerei printing house, which the ministry oversees, and shell companies in Panama.
Earlier this week the finance ministry denied it had ignored warnings from an informant in South America.
Last Sunday, Schaeuble urged countries to work together in the fight against tax cheats and money launderers by sharing national lists naming the beneficiaries of shell companies while warning that recalcitrant countries could be blacklisted.