Republican National Committee chairman Reince Priebus on Sunday warned there could be consequences for former presidential candidates who don't endorse their party's nominee.

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"People who agreed to support the nominee, that 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 on CBS's "Face The Nation."

"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 that we're going to evaluate the process – of the nomination process and I don't think it's going to be that easy for them."

Republican candidates signed a pledge during the primary season stating they would support their party's eventual nominee.

Host John Dickerson questioned whether Kasich — who has said it's unlikely he'll cast his vote for Trump — would be penalized if he wanted to run again.

"People in our party are talking about what we're going to do about this. I mean 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, no matter who that person is," Priebus said.

"So what's the penalty for that? It's not a threat, but that's just the question that we have a process in place," he continued.

"And if a private entity puts forward a process and has agreement with the participants in that process, and those participants don't follow through with the promises that they made in that process, what-- what should a private party do about that if those same people come around in four or eight years?"