Roy Hodgson has worked closely with Scandinavian players during his career and recently managed Norwegian club Viking Stavanger.
Carew, 30, has a year left on his contract but looks certain to replace compatriot Erik Nevland, who has been told he can leave the Europa League semi-finalists in the close season.
The former Valencia striker, who has scored 35 goals in nearly 100 appearances for Villa, would attract a bid of around £3.5m, and make an ideal partner for Bobby Zamora.