Insiders told the newspaper the Republican former U.S. senator from Tennessee tops a short list to take over Harvey's duties when the 87-year-old Chicagoan steps down.
Harvey attracts 17 million listeners a week to his three daily "news and comment" programs, the Post said. He is currently in the middle of a 10-year, $100 million contract.
A Thompson rep referred the Post to ABC Radio for comment.
"We have spoken to Senator Thompson, but we talk to talent all the time about many different roles," the ABC spokesman said. "There's no replacing Paul Harvey."