(CNN) Donald Trump may be finally gearing up to do what many Republican leaders have hoped: soften his rhetoric and pivot to the center.

He hasn't done that yet. But there are growing signs that the presumptive Republican nominee is aiming to make his campaign more palatable to a general election audience.

His campaign is putting the finishing touches on a policy memo that would change his proposed ban on Muslim immigration to the United States. Instead of focusing the ban on Muslims, Trump would ban immigrants coming from countries with known terrorism links, training and equipment.

Meanwhile, he's eased off his hardline language calling for deporting all undocumented immigrants living in the U.S.

The new tone follows several tough weeks for the Trump campaign as it has confronted a series of negative headlines that raised questions about his treatment of women and minorities and his temperament to be president. The big question is whether the softer tone is a blip that will soon be replaced by the familiar tough talk -- or a longer-term strategy that could help him take on Hillary Clinton.

Muslim ban, illegal immigration

Trump first proposed a ban on Muslim immigrants in the aftermath of the terror attack in San Bernardino, California, last year.

In the wake of this month's shooting at an Orlando gay nightclub, Trump has focused on the need to ban Muslims from "terror states" -- instead of all Muslims. He even hinted at the possibility of allowing in Muslims from those countries as long as they are "even more strongly vetted."

"People coming from the terror states -- and you know who I'm talking about when I talk about the terror states -- we are going to be so vigilant you wouldn't believe it and frankly a lot will be banned," Trump told CNN Saturday when pressed on where his Muslim ban proposal stands.

Minutes earlier, Trump had struck a less defiant tone when asked by reporters whether as president he would allow a Scottish Muslim into the U.S.: "Wouldn't bother me, wouldn't bother me," Trump said.

Trump's campaign, though, was eager to paint the billionaire's position in a less controversial light.

"It is about terrorism and not about religion," said Trump's national finance chairman Steven Mnuchin, who was traveling with Trump on a tour in Scotland.

Pressed for clarity on Trump's position , his spokeswoman Hope Hicks confirmed to CNN Saturday that Trump wants the Muslim ban to focus on those coming to the U.S. from the still undesignated "terror states," marking the campaign's first acknowledgment of an apparent pivot on the controversial policy.

Trump this weekend also rebuffed characterizing his plan to deport all estimated 11 million undocumented immigrants living in the U.S. as "mass deportation."

"I would not call it mass deportations," Trump told Bloomberg Politics on Saturday.

Instead, he said: "We are going to get rid of a lot of bad dudes who are here...that I can tell you."

And rather than emphasizing the criminal character of undocumented immigrants as he has in the year since announcing his presidential bid and calling Mexican immigrants "rapists," Trump stressed that his immigration policies would have "heart."

A Trump adviser told CNN Tuesday that the ban proposed in December was "a policy warning about the threat of seeing the refugees being infiltrated by ISIS or jihadists. Since then, ISIS has claimed it would infiltrate them validating Mr. Trump's concerns."

"The general ban statement was a suggestion to start a discussion about the threat, and indeed the debate started," the adviser added. "The ban concept naturally started to evolve into narrower categories and more precise areas. The principle of a ban on any force that would threaten the American people, either directly or by embedding itself into larger groups, is firm. Strategies to detect the threat will help address the particular penetration."

But Trump's proposal to ban all foreign Muslims was not simply a "warning" about the potential threat of refugees. Trump had previously already called for the U.S to stop accepting Syrian refugees and separately called on the U.S. to ban all foreign Muslims in response to the San Bernardino terror attack.

Why now?

The hints at a softening follow Trump's decision last week to fire controversial campaign manager Corey Lewandowski, who conceived and embodied the campaign's mantra, "Let Trump be Trump." The shakeup put Trump's campaign chairman and chief strategist Paul Manafort -- who for months has hinted at and encouraged tone and policy shifts -- more fully in charge.

Manafort, a veteran Republican strategist with deep ties to the party establishment, had battled for months with Lewandowski over the need for Trump to adjust his rhetoric and pivot to the general election.

The changing of the guard immediately began to herald a more traditional political operation as the campaign quickly amped up the pace of rapid response and fundraising emails. And in a prepared speech from Trump last week, the de facto GOP nominee laid out his most concise and well-argued case against Clinton to date.

Manafort's goal of triggering a rhetorical downshift seemed to find its legs in Trump's markedly less inflammatory response last week to the Supreme Court's ruling that blocked President Barack Obama's executive action, which would have shielded millions of undocumented immigrants from deportation.

In a statement, he didn't tout the need to build a wall or call for the deportation of millions of undocumented immigrants. Instead, it simply reinforced Trump's broader message about law and order, national security and executive overreach.

"The executive amnesty from President Obama wiped away the immigration rules written by Congress, giving work permits and entitlement benefits to people illegally in the country. This split decision also makes clear what is at stake in November," Trump said in the statement. "It is time to protect our country."

Until Lewandowski's dismissal, Trump showed few signs of changing, squandering six valuable weeks when Clinton was still fighting to clinch her party's nomination. Just as he continued to relive his greatest hits against fellow Republicans in the primary, Trump also doubled down on some of his most controversial proposals, like his call for surveillance of U.S. mosques.

But Trump must now confront the bleak reality that his doubling down on his controversial policies and rhetoric aren't giving him a boost with the general electorate.

The most recent polls show Trump trailing Clinton by anywhere from five to 12 points. And Trump's favorability numbers haven't improved, as about 6-in-10 Americans have an unfavorable view of Trump, according to the most recent polls.

Walking the tightrope

Under Manafort's guidance -- and practically unfettered influence -- Trump's campaign has eagerly worked to smooth over GOP establishment concerns and present Trump as a more presidential, and less controversial, candidate.

But as Manafort's own past efforts have proved, toning down Trump isn't easy.

When Manafort told a closed-door meeting of RNC officials earlier this year that Trump would be "evolving" from "the part that he's been playing" in the primaries, Trump emerged on stage at a rally the next day as brash and defiant as ever. And when Manafort teased a shift on Trump's Muslim ban proposal, Trump bristled at the suggestion and said his adviser must have been misquoted.

Trump, who has said he places a high value on loyalty, has been wary of giving his supporters any inkling of betrayal.

While his campaign aides affirmed on Saturday that Trump does not want to ban all foreign Muslims from coming into the U.S., the billionaire was more hesitant and declined in an interview with CNN the same day to disavow the controversial policy that has been a hallmark of his campaign platform since December.

And after telling Bloomberg that he wouldn't refer to his plan as "mass deportation," he tweeted that he didn't like the term, but emphasized that "we must enforce the laws of the land."

Trump has also spent little time in the last week rallying fervent supporters while standing on a teleprompter-less stage.

Instead, Trump's schedule included a speech delivered with prepared remarks and a slew of fundraisers with establishment financiers aimed at jumpstarting his struggling fundraising operation.

If Trump is changing, though, he's sticking to his own script and continuing to emphasize that he won't change.

"I do what I do," Trump told NBC News on Monday. "I don't care. I do me."

Earlier on Monday, some speculated that a softer tone was in the works when Trump didn't call Sen. Elizabeth Warren "Pocahontas" in a tweet once again knocking the Massachusetts Democrat for claiming Native American roots.

But in his phone call with an NBC reporter later on Monday, Trump unleashed on Warren.

He called her a "fraud" and "very racist." And he also called her "Pocahontas."