That unity is about to be tested as debate begins in the Senate Monday evening on immigration - specifically what to do about the so-called "dreamers" - immigrants brought to this country illegally as children.
The Senate did not vote to open the debate until Monday evening, and held no floor debate after that. Another 1.1 million young immigrants are eligible for Deferred Action on Childhood Arrivals (DACA) but did not apply.
Mr Porter left his job in the West Wing last week, but questions have persisted about how long senior White House officials knew about the allegations and why Mr Porter was given access to sensitive information without permanent security clearance.
He stressed that "there's no secret plan to push this in any direction".More news: McDonald's nixes cheeseburger from Happy Meal as part of larger health push
The proposal set forth by the GOP members is in line with President Donald Trump's four-point immigration plan.
But a major variable hangs over the efforts of lawmakers as they begin what could be a weekslong and politically bruising process: What kind of legislation will President Trump ultimately support? John McCain, R-Ariz., out for cancer treatment.
And President Donald Trump has been clear that he doesn't want anything short of a four-pillar solution- one that includes border security, protections for DACA recipients, an end to the diversity lottery and a massive overhaul of the country's legal immigration system.
Few expect Trump's plan to attract 60 votes, but Rounds said he believed the moderates' proposal could. "As soon as you add in a path to citizenship, you have to deal with chain [migration] and a lot of other things".More news: Kanye West Settles $10 Million Insurance Claim with Lloyd's of London
While the discussion at the McConnell Center was filled primarily with pleasantries - Schumer called the Kentucky Republican his "friend" numerous times - the minority leader also acknowledged that the Senate faces a complicated week, including the immigration issue.
It was unclear how their package will be received by Republican and Democratic leaders. Many Republicans are in the same boat.
And the senators called for much stricter limits on extended family who can be sponsored for future immigration - though parents would still be allowed to come on generous visitor visas. "Rather, this is the moment for a narrow bill", the New York Democrat said. McConnell indicated on Tuesday that the debate would be concluded at the end of the week.
In the House, Speaker Paul Ryan is taking fire from some in his rank-and-file who want a broader bill that can better enable a crackdown on border crossings and employers who hire undocumented immigrants.
In a statement released by the White House, Trump urged the Senate to support legislation offered by Republican Senator Chuck Grassley that would also scale back two immigration programs that bring more than 300,000 people into the United States each year.More news: Alzheimer's disease reversed in mouse model