The "Colbert Report" host "has indicated that he's willing to take over the Late Show when the time comes," according to people familiar with discussions. He would be leaving behind his "conservative" persona should he take the helm at CBS.
The network has also discussed the matter with with Colbert's Comedy Central colleague, Jon Stewart, according to the report.
Letterman announced he would be stepping down on his show Thursday.
“We don’t have the timing of this precisely down; it will be at least a year or so," he said. "But at some time in the not-too-distant future — 2015, for the love of God."
For a snippet of Stephen Colbert out of character, check out some footage of him introducing his sister who ran for a congressional seat in South Carolina.