Republican National Committee Chair Reince Priebus suggested Sunday that former GOP presidential candidates might be penalized if they refuse to back Donald Trump's White House bid.

"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 ... those people need to get on board," Priebus said on CBS' "Face the Nation."

"If they're thinking they're going to run again someday," Priebus added, "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."

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Declared GOP candidates, Trump included, signed a loyalty pledge last year to support whoever became the eventual nominee. The pledge has been a perennial source of tension throughout the election cycle. Some Republicans once feared that Trump would launch a third party run if he lost the nomination and siphon enough votes to assure a Democratic victory.

Related: Priebus 'Apoplectic' Over Trump's Refusal to Endorse Paul Ryan

When asked if Trump's primary opponents who do not make good on the pledge will be punished by the GOP, Priebus contended that "these are things that our party is going to look at."

Certain high-profile Republicans have made public rejections of Trump and his campaign tactics.

Gov. John Kasich and Sen. Ted Cruz, who formed an unsuccessful alliance against the businessman last spring, made headlines during the Cleveland convention for rebuffing the Trump campaign. Kasich did not speak at the convention although it was held in his home state of Ohio. Sen. Ted Cruz defiantly took to the stage and told jeering convention-goers to "vote your conscience."

Former Presidents George H.W. Bush and George W. Bush did not attend the Republican National Convention and declined to endorse Trump. Former Gov. Mitt Romney, who was the GOP nominee in 2012, has harshly criticized Trump, saying he appeals "to racism, xenophobia, misogyny, violence."

Other notable holdouts include former Gov. Jeb Bush, Sen. Lindsey Graham, Sen. Ben Sasse, Rep. Adam Kinzinger, Sen. Susan Collins and Sen. Jeff Flake.

However, several Republicans -- even some who have previously been critical of the businessman -- have tepidly coalesced around him, arguing that a Trump presidency would be more conducive than a Clinton presidency in advancing their interests. House Speaker Paul Ryan, former Hewlett-Packard CEO Carly Fiorina, Senate Majority Leader Mitch McConnell, Sen. John McCain and Sen. Marco Rubio are among this group.

Priebus insisted that his hints at potential consequences for refused endorsements were "not a threat."

“People in our party are talking about what we’re going to do about this," Priebus said. "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, it's just the question that we have a process in place ... what should a private party do about that if those same people come around in four or eight years?”