President Trump said Thursday he's "fairly close" to a deal with Congress that would protect young undocumented immigrants who came into the country illegally as children and address border security.

But even as Trump signaled progress on a grand bargain on immigration, he also disputed claims by Democratic leaders that he had agreed to drop his insistence that Congress eventually pay for a wall along the length of the Mexican border.

"The wall will come later, we're right now renovating large sections of wall, massive sections, making it brand new," Trump told reporters as he departed the White House for Florida, where he was to tour hurricane damage.

But his first priority, he said, was forging a deal that would combine "massive border controls" with some kind of legal status for so-called DREAMers, the children of undocumented immigrants.

President Obama had suspended deportations of DREAMers under a program known as DACA, or Deferred Action for Childhood Arrivals. But the Trump administration canceled that program this month, giving a six-month grace period in order to give Congress time to decide their legal status.

"We're working on a plan, subject to getting massive border controls. We're working on a plan for DACA. People want to see that happen. You have 800,000 young people, brought here, no fault of their own. So we're working on a plan, we'll see how it works out. We're going to get massive border security as part of that," Trump said. "And I think something can happen, we'll see what happens, but something will happen."

News of a possible breakthrough came Wednesday night after Trump dined with House Rep. Nancy Pelosi and Sen. Charles Schumer, the top Democrats in the House and Senate. Afterward, the Democratic leaders said they had a "very productive meeting" on the immigration issue.

Pelosi said Thursday that Democrats thought they had an agreement to incorporate the DREAM Act into the immigration package, with the border wall dealt with separately.

The DREAM Act is a bipartisan bill that would not just protect the DACA recipients from deportation, but also give them a way to earn citizenship. “I do believe there is an understanding that down the road there is an eventual path to citizenship,” she said.

But asked Thursday if he favors amnesty for undocumented immigrants Thursday, Trump shouted back to reporters, "The word is DACA."

The negotiations mark the second time in as many weeks that Trump — a real estate mogul who campaigned on his deal-making prowess — appeared to bypass congressional Republicans to deal directly with Democratic leaders. Last week, Trump agreed to a budget package that included a suspension of the debt limit, short-term spending and $15 billion in hurricane relief — to the consternation of some Republicans who felt he gave up too much leverage.

But this time, Trump said he spoke with House Speaker Paul Ryan, and Senate Majority Leader Mitch McConnell, and the Republicans are "very much on board."

House conservatives had a mixed reaction to the development. Rep. Mark Meadows, R-N.C., said border security has always been a broader issue than the wall. "You can tunnel under a wall," he said.

"Obviously the Trump voter is not in favor of amnesty," he said. But most undocumented immigrants aren't seeking citizenship, he said — just the ability to work without fear of deportation.

"It's a very complex issue that we're not going to solve in one dinner with two Democrats," said Meadows, chairman of the House Freedom Caucus.

Given the differing accounts of the meeting, it's unclear how close the two sides are to an actual agreement.

"No deal was made last night on DACA," Trump said in an early morning tweet. "Massive border security would have to be agreed to in exchange for consent. Would be subject to vote."

Trump hopes to use the so-called DREAMers as leverage to fulfill his campaign promises for tougher border security and immigration enforcement. But while Attorney General Jeff Sessions said the president doesn't have the power to defer deportations without approval from Congress, Trump has also tried to show sympathy for the DREAMers.

"Does anybody really want to throw out good, educated and accomplished young people who have jobs, some serving in the military?" he tweeted Thursday, adding: "They have been in our country for many years through no fault of their own."

– Eliza Collins and Deirdre Shesgreen contributed from Washington. David Jackson contributed from Joint Base Andrews, Md.