Clinton spoke with Dan Rather of CBS News in an interview that aired Tuesday night on "60 Minutes II."
"You guys make more money than I have," Clinton joked. "Maybe it's not a bad idea. You know, I hear it costs a lot of money to support a senator."
NBC said it had discussed a possible Clinton show with Harry Thomason, a friend of the president who produces television shows with his wife, Linda Bloodworth-Thomason.
But a White House spokesman said the president "had no plans to do such a show" and had not yet confirmed his plans.
During the Rather interview, Clinton did rule out running for mayor of New York City, governor of Arkansas, governor of New York and any other public office at least for now.
"I need to take a couple of months to just go down. I need some rest. I've been working like crazy for 27 years," Clinton said. "I want to have some time to rest and just be a private citizen again."
In a wide-ranging conversation, Clinton reflected on his "finest hour" as the 42nd president of the United States.
"I think when we prevailed in both houses by one vote on the economic plan in '93, that's what really turned the economy around and made possible so much else that happened."
The record-setting economy has coincided with most of his presidency, giving him reason to be optimistic for the country's economic future.
"I think that we've got quite a bit of life left in the economy, and the American people are still working hard, and they are very innovative. I expect them to have a good year next year," Clinton said.
The president discussed what he called one of his darkest days of his presidency, the day 18 U.S. soldiers were killed in Somalia during U.N. military sweeps in 1993.
"It was awful because of the circumstances, which I hope to be able to talk about in some detail some day. But to lose them all, in what was a humanitarian mission … it was the dark day."
Clinton refused to categorize the impeachment vote against him as a low point, saying "by the time they got around to voting, I knew what was going to happen. My darkest day came long before that when I had to come to terms with the fact that, you know, I made a terrible personal mistake, which I tried to correct in private, then got dragged into public. That was dark for me."
Clinton described his affair with former White House intern Monica Lewinsky "as a sad chapter in my life that I wish were not public."
When asked what would his wife, Sen.-elect Hillary Clinton, write about him in her multimillion-dollar book deal, he chuckled, "I don't know if there's $8 million worth to say. You all know it already."
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