Wilder's handlers however are adamant that they believe a return with Fury, who came back from a devastating 12th round knockdown to hold on for a draw, is the more attractive fight.
The champion scored two knockdowns in the contest but struggled throughout to find the target against the tricky and tactically astute Fury, who used his superior boxing skills to control proceedings for long spells.
When asked if this was his preference, Fury was vague at post-fight press conference.
The three judges were divided on the outcome, with one scoring it 115-111 for Wilder, another 114-110 for Fury and the third 113-113.
Deontay Wilder still hopes to face Anthony Joshua in a heavyweight unification bout but is happy to hand rival Tyson Fury a rematch following their epic draw. It doesn't matter to me where we do it. I was too hesitant. I think my emotions got the best of me and I don' think, I know, my emotions got the best of me just wanting to just really get him out of there. I went to Germany to fight [Wladimir] Klitschko, and I went to America to fight Deontay Wilder.
"No wonder AJ didn't want no part of that right hand!"
With just two minutes left in the fight, Wilder buckled Fury's knees with a right hand and knocked him senseless with a ideal left on the way down. American Hurd got his back off the ropes to land a crippling body shot.
Pedro to the rescue
Staples Center had a frenzied atmosphere after the high-energy introductions, but the fighters settled into a technical bout early.
The opening rounds were to Fury's liking.
Fury was elusive and creative in the seventh and eighth rounds, and Wilder appeared to be out of answers.
Wilder finally found a home for the right hand in the ninth round.
The punch wasn't the biggest of the fight, but Fury was stunned - and he responded by getting up and raising his aggression in an exciting round.
"To take something away from someone who has come from hell and back, to ruin the biggest comeback in boxing history, probably in sports history, is a disgrace", raged Fury's trainer Ben Davison. He never thought he'd box again.
"Well the future of boxing is brighter because the heavyweight division right now is so vibrant".
The 10-8 round proved crucial in the final reckoning, though.