Washington (CNN) One of Donald Trump's top supporters insisted Sunday that the Republican nominee is backing away from one of his most controversial immigration proposals: mass deportations.

In an interview with CNN's Jake Tapper on "State of the Union," former New York Mayor Rudy Giuliani said Trump doesn't want to break up immigrant families in America.

Giuliani told CNN's Jake Tapper on "State of the Union" that Trump "would find it very, very difficult to throw out a family that has been here for 15 years and they have three children, two of whom are citizens. That is not the kind of America he wants."

Giuliani's comments follow a hardline immigration speech Trump delivered last week in Arizona. The former mayor cited an Associated Press report characterizing Trump's comments as "a sharp retreat" from his pledge during the primary season to remove all 11 million undocumented immigrants from the United States.

Still, other Republicans, including Arizona Sen. Jeff Flake, rejected the idea that Trump had softened. And Trump's running mate, Mike Pence, and campaign manager Kellyanne Conway were coy on the issue. Their comments triggered criticism from the campaign of Democratic nominee Hillary Clinton, over "attempts to mislead" about Trump's immigration plan.

Trump's Phoenix speech took a tough line on his signature issue of immigration. He rolled out a list of 10 proposals to crack down on illegal immigration, including building a wall along the US-Mexico border and ramping up deportations of undocumented immigrants who have committed other crimes within the United States.

"In a Trump administration, all immigration laws will be enforced," Trump said. "As with any law enforcement activity, we will set priorities. But, unlike this administration, no one will be immune or exempt from enforcement -- and ICE and Border Patrol officers will be allowed to do their jobs. Anyone who has entered the United States illegally is subject to deportation -- that is what it means to have laws and to have a country."

Trump in his speech drew a line at any form of amnesty.

"For those here today illegally who are seeking legal status, they will have one route and only one route: to return home and apply for re-entry under the rules of the new legal immigration system that I have outlined above. Those who have left to seek entry under this new system will not be awarded surplus visas, but will have to enter under the immigration caps or limits that will be established."

Trump added: "We will break the cycle of amnesty and illegal immigration. There will be no amnesty."

But on Sunday Giuliani insisted "all 10 policies are largely directed toward criminal illegal immigrants," drawing a distinction between those immigrants and others who haven't broken additional laws in the country.

Giuliani said Trump was calling for the United States to secure its borders and remove those who have committed additional crimes -- and only then address what to do with the majority of undocumented immigrants living in the United States, including so-called "Dreamers" -- undocumented immigrants brought into the country as children.

"That's the point that he was making in the speech, and I agree with you that point got lost to some extent in the emotion of the moment," Giuliani told CNN's Jake Tapper on "State of the Union" Sunday.

He chalked that emotion up to the parents of Americans slain by undocumented immigrants who shared the stage with Trump, underscoring Trump's call to end so-called "sanctuary cities" that don't deport undocumented immigrants.

Democrats said they weren't buying Giuliani's claims about Trump's stance on immigration.

"Donald Trump should have gone to the Olympics as a gymnast, because there's a lot of contortionism there," said Labor Secretary Tom Perez, a Clinton supporter, on "State of the Union."

He cited Trump's use of a Phoenix-area sheriff known for his sharp crackdown on undocumented immigrants on-stage last week.

"If you're trying to get kinder and gentler, it's a curious strategy to get Joe Arpaio to introduce you at a speech," Perez said. "He stands for what Donald Trump stands for, and that is, Mexicans are 'rapists.'"

Sen. Flake, who isn't yet supporting Trump, told Tapper on "State of the Union" that he didn't get the same message as Giuliani from Trump's speech.

"If it was there, it was buried pretty deep. And no, I didn't catch it," he said of any new indications of Trump's openness to allowing some undocumented immigrants to remain in the United States.

Of Trump's speech, Flake said: "It just really doubled down on a lot of the rhetoric that he used before, and it really didn't explain with any clarity where he's going to move ahead in the future."

Flake's comments caught the attention of Trump. On Sunday night he tweeted, "The Republican Party needs strong and committed leaders, not weak people such as @JeffFlake, if it is going to stop illegal immigration."

The Republican Party needs strong and committed leaders, not weak people such as @JeffFlake, if it is going to stop illegal immigration. — Donald J. Trump (@realDonaldTrump) September 4, 2016

Trump sooner followed up on Twitter with a more personal jab at Flake.

"The Great State of Arizona, where I just had a massive rally (amazing people), has a very weak and ineffective Senator, Jeff Flake. Sad!"

The Great State of Arizona, where I just had a massive rally (amazing people), has a very weak and ineffective Senator, Jeff Flake. Sad! — Donald J. Trump (@realDonaldTrump) September 4, 2016

Differing views on immigration among Team Trump members

Trump's vice presidential pick, Gov. Pence, on Sunday deflected questions about the fate of those undocumented immigrants who haven't committed additional crimes in the United States, on NBC's "Meet the Press."

"With all due respect to the media's focus on the 11 million or whatever that number is, he was focused on the more than 300 million people who are citizens of this country and are here legally in this country, and driving policies in immigration that will work for them, work for the future of our nation," Pence said.

Trump campaign manager Kellyanne Conway also wouldn't say what Trump's eventual plan for the majority of undocumented immigrants living in the United States would be.

On ABC's "This Week," Conway said that "once you enforce the law, once you get rid of the criminals, once you triple the number of ICE agents, once you secure the southern border, once you turn off the jobs magnet, jobs and benefit magnet, then we'll see where we are."

"And we don't know where we'll be. We don't know who will be left. We don't know where they live, who they are. That's the whole point here, that we've actually never tried this," Conway said. "He will rescind all those executive amnesties and try to work with the Congress. And so at least he's trying to solve a problem."

The comments by Conway and Pence brought criticism from the Clinton campaign.

"Donald Trump has stated very clearly throughout his campaign that he will deport everyone who is undocumented, something that was reinforced in his speech in Arizona last Wednesday," Clinton campaign manager Robby Mook said in statement Sunday afternoon.

"What we saw today is Mike Pence and Trump's top campaign officials attempt to mislead voters about their mass deportation policy by using soft words to describe harsh tactics -- one of the oldest tricks in the book," Mook added. "Immigrant families know the meanings of 'humane' and 'fair' and can see straight through their cynical ploys. Trump's message to immigrant families is clear: everyone must go."