The Express & Star says the 24-year-old joined the Canaries on a six-month deal in August and has drummed up a partnership with Elliott Ward, which has helped the team to fourth in the Championship.
Barnett has practically no future at the Hawthorns and Norwich manager Paul Lambert is hoping a deal can be struck to keep him at the club.
He said: "The way he has been playing with Elliott Ward is fantastic.
"Leon is here on loan but if we could do something to keep him then, absolutely, I would look to keep the lad.
"Sometimes when you get a loan you just don't know if they really care, but Leon has been colossal for us.
"If we can do something then I will certainly speak to him about it."