The simmering tensions between the GOP and those Republicans refusing to support Donald Trump for president exploded Sunday after party chairman Reince Priebus suggested those candidates may be barred from running in the future.

"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." Priebus said on CBS' "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."

Priebus' comments were seen as a rebuke of two of Trump's rivals for the nomination, Texas Sen. Ted Cruz and Ohio Gov. John Kasich, who have refused to endorse Trump and have often spoken out against him.

Kasich, Cruz and the 14 other GOP candidates for the White House pledged last year to support the eventual nominee. And while Kasich has not openly campaigned against Trump, his camp issued a blistering response to Priebus' comments Sunday night, blaming the GOP chairman for clearing the way for Trump to destroy the party.

"The idea of a greater purpose beyond oneself may be alien to political party bosses like Reince Priebus, but it is at the center of everything Governor Kasich does," said spokesman John Weaver in a statement. "He will not be bullied by a Kenosha political operative that is unable to stand up for core principles or beliefs."

"In fact, Reince should be thanking the governor for standing for an inclusive, conservative vision that can actually win a national election and improve our country," Weaver continued. "The governor is traveling the nation supporting down-ballot Republicans and preventing a potential national wipeout from occurring on Reince's watch."

Party officials shrugged off the insults.

"We are totally focused on winning back the White House and maintaining our majorities in the House and Senate," RNC spokesman Sean Spicer said in a statement.

But the blowup comes as Trump has stumbled out of one of the best stretches of his campaign. Last week, he was able to capitalize on lingering questions about Democrat Hillary Clinton's health and family charitable foundation before he regained the negative spotlight by reviving the so-called birther issue surrounding debunked doubts about President Barack Obama's citizenship by claiming he had actually resolved the issue.

Trump has pulled even with Clinton in recent polls in Ohio, a state he must win if he hopes to win the election.

And Kasich may hold the keys to Trump's victory – or defeat – there.