"The time is right for an immigration bill as long as there is compromise on both sides," Trump told reporters at the White House.

Congressional Republicans were not ready for this.

"I wish they would send me a memo when they're going to do something because some of us are in meetings all day long trying to do our job, and then I come out and get asked 'what's your opinion,'" Arizona Republican Rep. David Schweikert told CNN.

Giving legal status to undocumented immigrants isn't necessarily a deal breaker in the House, but some conservatives argued that talking about legalizing millions before further action is taken on border security is premature.

"The most important thing is secure the border. Until that's done, you cannot even get to the other questions," Ohio Republican Jim Jordan, a member of the House Freedom Caucus, told CNN. "We have to demonstrate that we're going to actually secure the border so we have to do that first than we can get to all the other questions."

The number three House Republican, Rep. Steve Scalise, told CNN he hoped Trump would double down on border security, but signaled he wasn't ready to embrace any potential compromise dealing with those undocumented workers in the US now.

"I want to see us secure the border -- that needs to be our top priority on immigration," Scalise said.

There is no doubt that adding immigration reform to an already exhaustive agenda on Capitol Hill that includes repealing Obamacare, enacting tax reform, and passing a major infrastructure bill could be a step too far for a deeply divided legislature, but a senior administration official told CNN that Trump had still hoped it could be accomplished in his first term.

It wasn't that long ago that Republicans and Democrats teamed up to pass an immigration bill out of the Senate, but ultimately it died in the House of Representatives where it never came up for a vote partially because of concerns among House Republicans that giving undocumented immigrants a path to citizenship was too generous. What the White House is proposing doesn't go that far, but it still could be problematic for a party who just won the House, the Senate and the White House on a tough-on-immigration message.

Some Republicans, including early Trump supporter Rep. Chris Collins, applauded Trump's openness to immigration reform.

Texas Republican Rep. Bill Flores praised the concept, telling CNN, "I am there -- that's where I am, that's where my district is and I've got a fairly conservative district so I'm fine with that."

Flores said he polled constituents at his own town hall recently and they overwhelmingly voted for a plan that would offer some type of legal status.

"President Trump is in a unique position by coming out and advocating something where I think most of the country is that he can silence what I say is the hard left and the hard right on this - not silence, but he can sort of bring a perfectly reasonable solution that's important so I'm glad he's doing it," Flores said.

Some closed-door talks

Behind closed doors, one GOP member said that discussions are already underway among a group of House Republicans to get a bipartisan immigration bill across the finish line.

Members were meeting in hopes that at some point, they could pick up and run with any appetite from the White House to pursue immigration reform. The member, who spoke on the condition of background so they could speak freely about the ongoing discussions, said they had not been in consultation with the White House, but that the news was welcome.

"There's been a lot of work done," the member said, noting that they have reached out to Democrats, but so far the meetings have just been with Republicans. "There's always been the issue from a lot of folks --particularly from the right -- that say 'look, if we had border security we can do more.' There was zero confidence with (President Barack) Obama and I understand why. He didn't want to do border security."