It's highly probable that Tom Buchanan could have felt motivated to kill Jay Gatsby. Tom had a history of abuse and had lost control of his temper and his behavior previously, which Daisy could bring up when it served her purposes.
Do you know why we left Chicago? I'm surprised that they didn't treat you to the story of that little spree.
Tom disliked Gatsby and made no secret of his disgust with Gatsby's pretensions and attempts to include himself in high society. Tom "made a small investigation of" Gatsby's affairs and found enough information to be a potential threat to Gatsby's efforts to present himself as a wealthy and upstanding individual.
Tom felt threatened by the relationship between Gatsby and Daisy. He recognized that he didn't have some of the refinements or sensual appeal that Gatsby presented, but Tom had other ways of insuring that Daisy remained his possession.
Finally, Tom assumed Gatsby had been the driver of the car that hit and killed Myrtle Wilson, his mistress. "He ran over Myrtle like you'd run over a dog and never even stopped his car."