The newspaper also reports that the top scorer's former club, FC Midtjylland, is entitled to 40 percent of the transfer fee through a resale clause.
The contract with Birmingham extends until 2031.
"From a financial perspective, this is our second-largest player transfer within Djurgården," says Djurgården's sports director Bosse Andersson in a statement.
21-year-old Priske finished second in the Allsvenskan scoring charts last year with 18 goals. He was awarded the prize for the league's best striker at the Allsvenskan Grand Prix.
Later on Tuesday, Djurgården announced that they had finalized a replacement for Priske. 24-year-old Norwegian Kristian Lien has been signed from Dutch club Groningen.
Djurgården finished fifth last season.
