Enzi's comments come a day after Cheney, daughter of former Vice President Dick Cheney, announced her candidacy for Enzi's seat.
In a video message, Cheney, 46, said it was time for "new generation of leaders to step up to the plate."
Enzi disputed that he is one of those leaders who needs to be replaced.
"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."
In the same interview, Enzi jabbed Cheney. The senator from Wyoming said she had not done the traditional meet-and-greet campaigning that he had done.
"She's never been to any of my listening sessions," Enzi continued. "I've done listening sessions in Wyoming where I actually listen to what the constituents have to say. I'm an old shoe salesman. I listen to the customer; I know who the customer is. I see how what fits with the inventory we've got, which is the legislation. I love to legislate."
Cheney announced her candidacy shortly after Enzi said he would run for reelection. Enzi said he was surprised by Cheney's decision, having known her family and being friendly with her father for years.
"She said that if I ran, she wasn't gong to run, but obviously that wasn't correct," Enzi said after he found out about Cheney's candidacy. "I thought we were friends."