Now the pressure is on Democrats to reject Trump's plan.
Trump's recent comments haven't helped bridge that gap.
"I would certainly advise against it", Kelly said.
Smith said that Trump talks a good game, but usually does not back up that with action.
Though Trump has repeatedly expressed an eagerness to broker major bipartisan legislation during his presidency, his rhetoric on the matter has lacked substantive follow-through. President Bill Clinton called for unity a month after he was impeached by the House of Representatives; President George W. Bush did the same when dozens of USA troops were being killed every month in Iraq; and President Barack Obama tried to inject a feeling of national healing after numerous school shootings.
DAVIS: There is still a lot of disagreement in the party over what the end goal is here. "They want to use it for an election issue, but it's now an election issue that will go to our benefit", Trump said. Instead, he said administration officials spoke of the need to fix and upgrade existing fencing, build additional barriers and make other improvements. And like a lot of Republicans, he's trying to make this case - it's not about the men and women of the Federal Bureau of Investigation.
"President Trump's idea of bipartisanship has been for Republicans to write a bill, introduce it, and then urge Democrats to support it".
Senate Majority Leader Mitch McConnell said this time he did not expect a repeat of the shutdown, for which both camps have traded blame. In 2009, the Brookings Institution, in tandem with Duke University's Kenan Center for Ethics, convened a bipartisan group, of which I was a member, to accomplish what the title of our report promised: "Breaking the Immigration Stalemate".
Congress will return this week to the political cauldron of immigration policy, and nearly every lawmaker will claim he or she is a player.More news: Former CIA Director Brennan says Nunes 'abused his office'
The U.S. Congress made no notable progress this week toward a deal on the status of 700,000 "Dreamer" immigrants, with President Donald Trump saying on Friday that one "could very well not happen" by a deadline next month.
Sanford noted that more than 30 House Republicans have now chose to retire, run for other office or outright resign amid controversy - hardly a resounding vote of confidence in the Trump administration.
Trump, whose election success hinged partly on his tough line on immigration, has said he is open to letting 1.8 million Dreamers stay in the country and eventually become citizens. "We're remaking the judiciary in a way that actually believes in upholding the Constitution", she said. "That's actually pretty courageous".
Trump reiterated his administration's offer to protect Dreamers - the young immigrants brought to the USA illegally by their parents at the center of Washington's immigration battles and the recent government shutdown - in exchange for wall funding and tightening of legal immigration laws.
Democrats want an assurance the Deferred Action for Childhood Arrivals program will be preserved as a condition of voting to maintain government funding.
That sentiment will be heightened as both parties near 2018, said Republican strategist Doug Heye, a CNN political commentator. "Incredibly irresponsible. It in no way reflects reality", said Rep. Beto O'Rourke of El Paso, a Democrat hoping to unseat Sen.
All 435 seats in the House and 34 seats in the 100-seat Senate are at stake in November's election. And every delay allows positions to harden and pushes debate on a politically polarizing issue - how to resolve the fate of some 1.8 million young people who arrived in the US illegally as children - closer to the campaign season for congressional elections. "'I really am not happy with the way it's going from the standpoint of the Democrats negotiating". And they were best friends, and everybody got along.
Even later, Trump tweeted that Pelosi and Senate Minority Leader Chuck Schumer "have to get moving fast, or they'll disappoint you again".
Some of the former "Gang of Eight" members who came up with a comprehensive immigration reform bill in 2013 have gotten back together and hashed out a compromise that addresses the "four pillars" the White House is calling for: Protections for the DREAMer population, some funding for border enforcement, and changes to the visa lottery and family-based immigration.