The boxer has faced criticism after he made comments on homosexuality and made sexist remarks about women.
He was cleared of hate crime allegation by police, Greater Manchester Police said that while the comments had been recorded as a "hate incident", it was taking no further action against the 27-year-old.
Fury is alleged to have compared homosexuality with paedophilia and joked about being violent to women amongst other remarks.
Gabby Logan has said that he should remain on the shortlist for BBC Sports Personality of the Year but said she would be "disappointed" if he were to win.
To date more than 139,000 people have a signed a petition calling for Fury to be removed from the competition.
Earlier in the month he defeated Wladimir Klitschko to become a world heavyweight champion.
Gabby Logan, who will present the awards show alongside Gary Lineker and Clare Balding, has said it was right for his sporting achievement to be recognised: "He's on the list because of his sport.
"That is the argument that the BBC has put forward, and we all buy into it because that's the truth of it. He's not on the list for his opinions, clearly.
"This whole idea that we just shut people up, that's not the way forward. I'd rather have the debate about why what he says, in my eyes, is wrong than tell him to shut up and go away.
"Because he's not going to stop thinking or believing that, and he's not going to stop spouting it, so why don't we have a conversation about it?" she told The Guardian.
But she added that she found his views "objectionable" and when asked how she would feel if he won the public vote, said: "I'd genuinely feel confused. Are people voting because they think his achievements from a sporting viewpoint are head and shoulders above the rest? So much so that they can ignore the stuff around him? I would be a bit disappointed. I'd think, 'Are people tying to be ironic?'."
Shadow culture secretary Michael Dugher has for him to be dropped from the list and sports minister Tracey Crouch said that as a sports star he is a role model to society.
The winner will be announced in a live programme on BBC1 on Sunday.
