The pessimism echoes similar comments they have made before. But the latest round of skepticism comes after a week in which Trump criticized leading figures in his party and the Muslim American parents of an Army captain killed in Iraq, highlighting the lingering concerns in the Republican Party as public polls show him falling badly behind Clinton.
Meanwhile, a pro-Trump super PAC released an ad hitting Clinton as it tries to help the business mogul rebound.
In an interview on CNN's "State of the Union," Kasich 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," Kasich said. "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."
Speaking on CBS's "Face the Nation," Flake cautioned that Clinton could win his state. Trump, the senator said, 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."
Arizona has gone for the Republican nominee in every election since 2000. It has gone for the Democratic nominee only once in the past 10 presidential elections.
"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."
The senator added: "Yes, I think he does have to change his positions and be a more serious candidate."
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 Atlantic 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.
Kasich has not endorsed Trump, and he skipped the Republican National Convention in Cleveland. He said he wants to see Trump embark toward a goal of "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. His initial refusal to do so angered many Republicans.
"As far as I'm concerned, McCain shouldn't even have to run for election in the Senate. He ought to be in the Senate as long as he wants to be," Kasich told CNN after praising the Arizona Republican senator.
Trump also drew criticism last week from a broad group of military luminaries when he criticized Khizr and Ghazala Khan, whose son Humayun Khan, an Army captain, was killed in Iraq in 2004. Trump said Khizr Khan had “no right” to go after him in his speech at the Democratic National Convention. The Khans support Clinton.
Meanwhile, pro-Trump super PAC Rebuilding America Now released an ad online raising questions about the financial ties between the Clinton Foundation and foreign entities and highlighting the Clintons' wealth.
Florida Gov. Rick Scott, a Trump supporter and national chairman of the super PAC, said on NBC's "Meet the Press" that the ad will begin running on TV in Ohio, North Carolina and Florida on Monday.
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 Khan family 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 "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.