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Two Republican officials who have been critical of Donald Trump raised new concerns about his ability to win in the key battleground states of Ohio and Arizona.

Speaking on CBS's "Face the Nation" on Sunday morning, Sen. Jeff Flake, R-Ariz., cautioned that Hillary Clinton could win his state - which has gone for the Republican nominee in every election since 2000 and has gone for the Democratic nominee only once in the past 10 presidential elections.

Trump, Flake cautioned, needs to take more "responsible positions" to hold on to a state that leans Republican but has threatened to become a more serious battleground because of a growing Hispanic population.

Asked whether Clinton could win Arizona, he said, "In 1996, Bill Clinton won Arizona. So, yes, it is possible."

"There's an increased urgency among Hispanics to vote, and if they do, it will be a changed ballgame here," Flake said. "We in Arizona realize that we've got to have meaningful immigration reform - you can't just throw out platitudes out there about a wall and Mexico paying for it and be taken seriously here."

"And so, yes, I think he does have to change his positions and be a more serious candidate," the senator added.

Flake has been critical of Trump in the past and on Sunday morning reiterated his broad concerns with the nominee's policy positions.

He said Trump needs to take different positions on immigration, his proposed ban on Muslim entry into the United States, his opposition to trade, and his calling into question the U.S. participation in the North American Treaty Organization.

"If none of us on the Republican side are pushing back and saying that needs to change, I don't think it will change," Flake said. "You can't go on and expect that you're going to be president of the United States when you make statements like that."

"We need a more responsible campaign, and we haven't seen it so far," he added.

Flake's comments were echoed by Ohio Republican Gov. John Kasich, who in an interview broadcast Sunday on CNN's "State of the Union" voiced serious doubts about Trump's ability to win the Buckeye State, which is a quadrennial battleground rich with electoral votes.

"He's going to win parts of Ohio where people are really hurting," said Kasich, a Trump critic. "There will be sections he will win because people are angry, frustrated and haven't heard any answers. But I still think it's difficult if you are dividing to be able to win in Ohio. I think it's really, really difficult."

Kasich has not endorsed Trump, and he skipped the Republican National Convention in Cleveland. He said he wants to see Trump embark on "unifying" voters.

The governor said he does not support Clinton. As for what he will do at the ballot box this fall, Kasich said he was undecided.

Trump did not appear on the Sunday morning news shows. On Twitter, he went after Baltimore Mayor Stephanie Rawlings-Blake, who defended Clinton and attacked Trump on CNN.

"I see where Mayor Stephanie Rawlings-Blake of Baltimore is pushing Crooked hard. Look at the job she has done in Baltimore. She is a joke!" Trump tweeted, referencing one of his nicknames for Clinton, "Crooked Hillary."

Several of Trump's surrogates appeared on the Sunday morning shows, including former New York mayor Rudolph W. Giuliani, who spoke on ABC's "This Week With George Stephanopoulos."

"I think his reaching out and supporting John McCain and Kelly Ayotte in particular, and Paul Ryan, who had been critical of him, you know, a couple of days earlier, shows that he has the ability and the understanding to realize that there are going to be disagreements and you've got to be able to reach out to the entire party," said Giuliani, referencing Trump's support for those Republicans after he initially refused to back them.

"I think you're going to see a lot more of that in his economic message," Giuliani added.

Seizing on Trump's week of controversies, Clinton's running mate, Sen. Tim Kaine, D-Va., noted that their campaign has stuck to an economic message, while Trump has been distracted by controversies of his own making.

Kaine noted that Trump spent much of the past week addressing criticism from the family of a fallen soldier, Capt. Humayun Khan, and explaining whether he would back Ryan's reelection campaign.

"Hillary and I have been on the road this week. We've been talking about jobs and the economy," Kaine said on NBC's "Meet the Press." "Donald Trump has been on the road talking about how the Khan family viciously attacked him and why he's not supporting the speaker in his own primary. How many different people does he want to fight against?

"Donald Trump is shadowboxing against every last person on the planet," he added.