"The nation as a whole is not going to elect the next president, 12 states are," Walker said in an interview with CNBC's John Harwood that was conducted in August but released on Tuesday. "Wisconsin's one of them. I'm sitting in another one right now, New Hampshire. There's going to be Colorado, where I was born, Iowa, where I lived, Ohio, Florida, a handful of other states. In total, it's about 11 or 12 states that are going elect the next president."

The presidential hopeful -- who has fallen recently in the polls — was answering a question from CNBC's John Harwood about whether he was too reliant on white voters, a group that carried his hero Ronald Reagan into office, but has been steadily shrinking.

Walker, who was considered a top-tier candidate when he launched his presidential bid in July, is suddenly having trouble in the polls, particularly in Iowa, a key state in his primary strategy. His fall has come as other candidates -- including retired neurosurgeon Ben Carson -- have had a spike in recent surveys.

Walker played up his success in winning over independent voters and crossover voters in 2012 -- people who voted for Walker in his summer recall election and Obama a few months later.

"Because what we found is many of those same Independents turned around and vote — 11% of them voted for Barack Obama in the fall," Walker said. "They were, oddly enough, Obama-Walker voters, which politically makes no sense. Why? Because they were people hungry for leadership. They were frustrated because they felt that Romney hadn't offered a clear plan of what he would do."