The 21-year-old impressed Tore Rismo, the club's director of football, when he starred for the Crewe reserves against Tromso in April.
"He played extremely well in that match against us and has had a kind of breakthrough in the English First Division this season," said Rismo.
Crewe, however, are unlikely to allow Hulse a permanent switch, and it looks as though the Norwegians will be forced to settle for the possibility of attracting the forward in a loan deal.
Neil Harrowell, the agent working on behalf of Tromso, confirmed the interest, saying: "The only thing I can say is that we are trying to find out if Crewe are willing to loan Hulse to us. My impression is that he is their number one choice."
A product of the much-vaunted Crewe youth academy, Hulse scored 12 goals in his 22 starts for The Railwaymen this season.
