Republican National Committee chair Reince Priebus said on Sunday that primary candidates who have not made good on their pledge to support Donald Trump could be punished by the party if they decide to run again in the future.

All of the Republican candidates for president signed a pledge last September promising to endorse the party's nominee "regardless of who it is." Some former candidates, like Ohio Gov. John Kasich and Republican Sen. Ted Cruz, have not formally endorsed Trump.

"I think these are things that our party's going to look at in the process," Priebus said on CBS News's Face the Nation when asked about potential penalties for those who did not follow through on the pledge. "And I think that people who gave us their word, used information from the RNC, should be on board. I mean if you want to take part in the process."

"People in our party are talking about what we're going to do about this," Priebus added. "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. 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, "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?"