Ohio Gov. John Kasich and former Florida Gov. Jeb Bush could be barred from seeking the Republican Party's presidential nomination in the future if they refuse to get on board with Donald Trump, party chairman Reince Priebus said Sunday.

"I think people need to get outside this Beltway and get on the road," Priebus said in an interview with CBS host John Dickerson. "This is probably one of the biggest movements as far as people across this country in modern history.

"Where we're at with the voters, one of the last polls I saw ... Republicans, I think it was about 91, 92 percent, we need to do a couple of percentage points better than we're doing as we move forward. Look, people who agreed to support the nominee, who took part in our process, they used tools from the RNC. They agreed to support the nominee. They took part in our process," Priebus said.

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"We're a private party. We're not a public entity. Those people need to get on board. And if they're thinking they're going to run again someday, I think we're going to evaluate the nomination process, and I don't think it's going to be that easy for them. I think these are things that our party's going to look at in the process. I think people who gave us their word, used information from the RNC, should get on board," he added.

Priebus said Republican voters were asking the RNC to take action against those who pledged to support the nominee but went back on their word, including Kasich and Bush.

"People in our party are talking about what we're going to do about this. There's a ballot access issue in South Carolina. In order to be on the ballot in South Carolina, you actually have to pledge your support to the nominee ... So what's the penalty for that? It's not a threat. It's just a question," Priebus said.