They can be hard to spot — let alone take at face value — amid the usual spray of insults and bombast, but every now and then President Trump emits a sensible idea.

Mr. Trump offered at least the appearance of such an idea when he strode into the White House briefing room late Wednesday and announced that he was open to a multiyear path to citizenship for the so-called Dreamers, about 700,000 immigrants brought to the United States illegally as children and granted protection from deportation by President Barack Obama. His words sent his surprised staff scrambling and heartened Democrats who have been seeking a just solution for these young people.

But the price of the offer became clear Thursday afternoon, when White House officials said that citizenship for the Dreamers, and about 1.1 million other young immigrants who could qualify for the status but didn’t apply, would come only as part of a bargain that included an additional $25 billion for border security and a crackdown on legal immigrants, including family members of those already in the United States.

Mr. Trump, who canceled the Dreamers’ protection in September, has professed his “love” for them, but the real policy is being made by hard-liners in the White House.