But Boehner also admitted it will be difficult to get anything done because Republicans don't trust President Obama:
"And frankly, one of the biggest obstacles we face is the one of trust. And the American people, including many of my members, don't trust that the reform that we're talking about will be implemented as it was intended to be. The president seems to change the health care law on a whim whenever he likes.
"And now he's running around the country telling everyone that he's going to keep acting on his own, keeps talking about his phone and his pen. And he's feeding more distrust about whether he's committed to the rule of law.
"Listen, there's widespread doubt about whether this administration can be trusted to enforce our laws. And it's going to be difficult to move any immigration legislation until that changes."
'Hide the cheese'
"I think what he's playing is 'hide the cheese,'" Democrat Party Chairwoman Debbie Wasserman Schultz said of Boehner a few hours after he spoke. "I mean, what they're doing here is just trying to delay and draw this out and pretend that they are committed to the principles of comprehensive immigration reform."
The Florida Democrat, appearing on MSNBC, accused Boehner of "making excuses" for not passing "comprehensive" immigration reform legislation.
"That sends a very clear message that they don't care about this issue; it's not a priority. And I think that's why we have enjoyed, in part, the support of Hispanic voters, and the Democratic Party will continue to because the Democratic Party is focused on the top priorities that are important to the Hispanic community..."
Boehner has insisted on a "step-by-step" approach to immigration reform, beginning with border security. "And the reason I said that we need a step-by-step common-sense approach to this is that so we can build trust with the American people that we're doing this the right way."
The Speaker said on Thursday that Republican leaders will continue to discuss immigration reform with rank-and-file Republicans. "But I think the president's going to have to demonstrate to the American people and to my colleagues that he can be trusted to enforce the law as it is written."
Boehner also noted that President Obama is asking Republicans "to move one of the biggest bills of his presidency, and yet he's shown very little willingness to work with us on the smallest of things." Boehner referred to four House-passed bills awaiting Senate action that address issues outlined by Obama in his State of the Union address.
"The president could reach out and work with us on those and begin the process of rebuilding the trust between the American people and his presidency."
Boehner said for 15 months, he's talked about the need for Congress and the president to work together on immigration reform: "It needs to get done. I'm going to continue to talk to my members about how to move forward, but the president is going to have to do his part as well."
White House spokesman Jay Carney said on Thursday that the Obama administration remains optimistic about the prospects for comprehensive immigration reform in 2014.
"We've seen significant movement among Republicans on this issue. And it is heartening to see that Republican leaders in Congress, including the Speaker of the House and others, identify immigration reform as a necessary priority. That's a good thing."
Carney said the challenges in getting a bill passed come from "within the Republican Party -- and they certainly don't have anything to do with the president."