Well, this goes beyond rumor. Jerry Jones has told NFL Network analyst and former Dallas Cowboys scout Gil Brandt that the team will not draft a quarterback with the No. 4 pick in the 2016 NFL Draft. Brandt said, "Jerry told me recently that he will unequivocally not take a quarterback at No. 4, where the Cowboys draft this year. I believe him."
Brandt raises some good points in terms of why he believes Jones. Brandt said that if the Cowboys think Tony Romo will make a full recovery this offseason, and if they think he's got at least three more years in him as the starter, the team needs to address more glaring needs with the top pick. This does not preclude Dallas from drafting a developmental QB option later in the draft, but the impact player makes more sense with that top pick.
The Cowboys defense has a few needs, with pass rusher likely prominent on the list. Greg Hardy showed some signs of his talent, but also was plenty problematic. Randy Gregory is dealing with a suspension to start the upcoming season. It's safe to say the Cowboys will be looking pass rush at some point. There are some intriguing options at No. 4, and it is also possible they could decide to deal down if the right offer comes their way.
Of course, this is also if you believe what Jones said. I think Brandt's reasoning for why he believes Jones makes sense. But things can change in a hurry.