The unsigned letter was sent to the Ministry of Culture complaining of Westerberg's leadership.
The newspaper reports it is unclear how many staff membesr are behind the letter. The Culture Ministry said it is looking into the matter and has contacted Westerberg.
"We take the allegations seriously," Ministry press secretary Catharina Henriksson said.
The museum's communications manager Christina Kåremo Sköldkvist said Westerberg denies that nepotism is behind his hiring practices.
"We had made a departure from the recruitment policy, but we felt we did the right thing," she told the newspaper.
The Natural History Museum in Stockholm is one of Sweden's largest museums, receiving some 500,000 visitors last year.