Washington (CNN) Donald Trump is declining to endorse a pair of Republican heavyweights -- House Speaker Paul Ryan and former presidential nominee John McCain -- in their primary fights, another sign of the fractures within the party.

Ryan and McCain face long-shot challengers in their respective primary races in Wisconsin and Arizona. Yet their party's presidential nominee would not bless them in an interview with The Washington Post on Tuesday.

"I like Paul, but these are horrible times for our country. We need very strong leadership. We need very, very strong leadership. And I'm just not quite there yet. I'm not quite there yet," Trump said of Ryan.

As for McCain -- a former naval aviator who was shot down and tortured during the Vietnam War -- Trump said he has "never been there with John McCain because I've always felt that he should have done a much better job for the vets."

Trump's public reprimanding of two of his party's most senior officials is a reminder that two weeks after the Republican National Convention, his party is far from unified. Both Ryan and McCain have endorsed Trump -- with Ryan spending significant political capital to do so -- yet Trump has not taken kindly to the criticisms launched by GOP elders.

Ryan faces Paul Nehlen in Wisconsin next week. Trump's phrasing -- "I'm not quite there yet" -- echoes comments Ryan made to CNN's Jake Tapper in May when he said he wasn't yet ready to back his party's standard-bearer.

Trump claimed in the Post interview that Ryan had sought his endorsement, which he was only giving "very serious consideration." Zack Roday, a Ryan campaign spokesman, quickly pushed back against Trump's claim, saying, "Neither Speaker Ryan nor anyone on his team has ever asked for Donald Trump's endorsement. And we are confident in a victory next week regardless."

A message left by CNN with the office of McCain, who faces a challenge from his right at the end of the month from former state Sen. Kelli Ward, was not immediately returned Tuesday afternoon.

But McCain's daughter, Meghan McCain, who has been critical of Trump in the past, simply tweeted, "Character. Matters."

Character. Matters. — Meghan McCain (@MeghanMcCain) August 2, 2016

Tough criticism for Trump

Both Ryan and McCain in recent days have sharply criticized Trump over his latest controversy, his disparaging remarks about the Muslim parents of Humayun Khan, a US Army captain who died in the Iraq War.

Ryan reiterated Sunday that he rejects Trump's call for a temporary ban on Muslim travel to the US after Trump took heat for criticizing Khan's parents.

McCain in particular was harsh, saying Trump did not have an "unfettered license to defame those who are the best among us."

The Arizona senator is meeting with Trump's running mate, Indiana Gov. Mike Pence, for a pre-scheduled meeting in Arizona on Tuesday.

Trump previewed some hostility toward Ryan in a tweet praising Nehlen on Monday after the businessman praised Trump and recently tweeted articles and comments supporting the GOP nominee.

"Thanks to @pnehlen for your kind words, very much appreciated," Trump tweeted.

Thanks to @pnehlen for your kind words, very much appreciated. — Donald J. Trump (@realDonaldTrump) August 1, 2016

In the Post interview, Trump also criticized New Hampshire Sen. Kelly Ayotte, who is in a tough re-election battle and hit Trump over his remarks against the Khan family. She is supporting Trump's candidacy but has tried to distance herself from him, claiming that she is not "endorsing" him.

"I call it like I see it and I'm always going to stand up for our military families and what's best for the people of New Hampshire," she tweeted after Trump's remarks were published.

I call it like I see it and I'm always going to stand up for our military families and what's best for the people of New Hampshire. — Kelly Ayotte (@KellyAyotte) August 2, 2016

Ohio Gov. John Kasich, a longtime Trump critic, rushed to the defense of Ryan, McCain and Ayotte, tweeting that they are "great leaders" who are "critical to holding our conservative majorities" in Congress.

President Barack Obama on Tuesday challenged Republicans over their support for their presidential nominee given his often-controversial rhetoric on the trail.

"What does this say about your party that this is your standard-bearer?" Obama asked of GOP leaders in remarks made at a White House news conference . "This isn't a situation where you have an episodic gaffe. This is daily and weekly where they are distancing themselves from statements he's making. There has to be a point at which you say, 'This is not somebody I can support for president of the United States, even if he purports to be a member of my party.' "

Obama said that denunciations from Republicans of Trump's remarks "ring hollow" without an accompanying withdrawal of support.

Could imperil Trump's support on Capitol Hill

One Republican congressman predicted that Trump's criticisms would backfire and harm the nominee rather than Ryan, given the speaker's support among House Republicans, many of whom are lukewarm about backing the controversial real estate mogul. The congressman, who has endorsed Trump but asked not to be identified in order to speak freely, said supporting Trump is now more difficult given his criticism of the Khan family.

Asked how Trump would work with Ryan on Capitol Hill if both men were to be elected without Trump endorsing the House speaker, campaign spokeswoman Katrina Pierson told CNN's Wolf Blitzer Tuesday that she would expect congressional Republicans to support the will of the people.

"Well, (Trump) would hope that any elected official, based upon the election, would understand that the policies that Donald Trump has been putting out -- considering he would have defied all odds, not just at the primary but in a general election as well -- and they would work to represent their people," Pierson said. "That's what this government is about. We are a republic, not a democracy. Sometimes people forget that."

As for Ayotte, who has been under fire from New Hampshire Democrats for being too close to Trump, state Republican Party committeeman Tom Rath said Trump attacking her "is a gift."

"Her numbers here are far better than his," Rath said, adding, "Independence is prized highly here and Trump just validated Kelly's authenticity on that."