The 2009 Man of Steel was offered a one year extension by the club, but Australian Hodgson wanted a two-year deal.
The stalemate has forced the Giants to tell Hodgson he is free to leave at the end of the current campaign.
Giants Managing Director Richard Thewlis said: "It was always our intention to see how Brett was going after the first two months of the season and as was the case last year his form is outstanding hence we offered him a contract extension for another year at the club.
"He wanted the guaranteed security of two years which we were not prepared to do at this stage and therefore gave his manager permission to explore the market elsewhere.
"His manager has informed the club that he can attain two years at other clubs and consequently we expect him to move on at the end of the year. He will leave with our very best wishes and thanks for his time here and will always be welcomed back in the future.
"I think its best that we get this situation out into the open immediately so our supporters know what's going on as these things usually leak out anyway and we have to look at our recruitment plans again for 2011 without Brett in the side."
For full story, read Sports Monday, free with Monday's Yorkshire Post newspaper.