Donald Trump’s mixed signals about easing his plan to deport all illegal immigrants are dividing his closest allies and prompting warnings he could lose core supporters if he abandons the signature issue of his campaign.

Even as some supporters said they would welcome a softer tone as a sign Mr. Trump is working to broaden his support, he said in a Thursday CNN interview both that it would be difficult to deport 11 million illegal immigrants and that he might do it anyway.

But in interviews this week with Fox News he backed away from his long-held proposal to deport the nation’s 11 million illegal immigrants, suggesting those without criminal records could stay if they pay “back taxes.”

That didn’t sit well with former Alaska Gov. Sarah Palin, who was among the first high-profile Republicans to endorse Mr. Trump in January. “If Mr. Trump were to go down a path of wishy-washy positions taken on things that the core foundation of his support has so appreciated, and that is respecting our Constitution and respecting law and order in America, then, yeah, there would be massive disappointment,” the party’s 2008 vice-presidential nominee said.

“Parts of the message we heard in the last week are clearly not consistent with the stringent position and message that supporters have received all along,” she said.