"It appears we've agreed to a one-year deal," Red Wings general manager Ken Holland told the newspaper.
Holland said Gustavsson is in Sweden and the deal will become official soon.
Gustavsson is coming off his best NHL season, going 16-5-4 with a 2.63 goals-against average and .907 save percentage as Jimmy Howard's backup.
Gustavsson started the final two games of the Red Wings' five-game loss to the Boston Bruins in the Eastern Conference First Round after Howard came down with the flu. He posted a 2.71 GAA and .917 save percentage despite losing both games.
Gustavsson, 29, has a record of 57-52-20 in 141 games with the Toronto Maple Leafs and Red Wings, with a 2.91 GAA and .900 save percentage.