Walton said in July he was stepping down from the post.
The Times said a deal with Zucker -- although not definite -- could be announced this week or early next week.
The former "Today" and current "Katie" producer left NBCUniversal nearly two years ago.
He couldn't be reached for comment Tuesday regarding a possible pact with CNN and his attorney did not return phone calls, the Times said.
Celebrity Families of 2014 [PHOTOS]