That said, automated surveys have traditionally proven less reliable than live calls, and the poll interviewed 422 Wyoming Republicans. The polling firm estimated a margin of error of nearly 5 percentage points.

In the video announcing her candidacy, Cheney said it was time for a "new generation of leaders to step up to the plate." 

Enzi, 69, has battled back against the insinuation that he is too old, saying Cheney hasn't put in the legwork necessary to represent Wyoming.

"I'm in really good health. I'm on the committees that I want to be on. I have seniority, which is really trust that you develop," Enzi told CNN. "If somebody new comes in, you don't trust them. I've developed trust on both sides of the aisle; people trust me that what I say I'm going to do, I do."