The Post said the offer is part of the network's plan for Couric after she leaves her anchor position at the CBS Evening News in June after five years.
Along with the "60 Minutes" gig, Couric would have to agree to serve as host of a daytime talk show, network insiders told the newspaper.
Couric's representative declined to comment on the story, the Post said Monday.
"Katie's on vacation this week," Couric's spokesman said. "Hopefully, the speculation about Katie can take a week off, too."
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