Fury's promoter Mick Hennessy has revealed that the 40-year-old rejected the offer despite previously claiming that he wanted a match to be arranged.
"I can't believe that Martin has turned down the offer to fight Tyson," said Hennessy, according to Sky Sports News. "He was given all he wanted - one of the best purses he's ever had, a chance to fight for both titles and fighting in his home city.
"In the end Martin talked a good fight with Tyson but that was all, he just didn't want to fight him and that's a shame."
Meanwhile, Canadian fighter Neven Pajkic has also reportedly turned down the opportunity to meet Fury, 22, in the ring.