McConnell previously promised Senate Democrats that he would bring up a bill on the chamber floor to address the pending expiration of the Deferred Action for Childhood Arrivals program if the government stays open past February 8. Ryan has said he will observe the so-called "Hastert Rule" of not calling any legislation that does not have the support of a majority of Republicans.
The White House's Office of Management and Budget "is now preparing for a lapse in appropriations", an OMB official said on condition of anonymity, calling on lawmakers to get the measure to President Donald Trump's desk "without delay".
The bill removes automatic spending cuts - known as sequestration caps - for both defense and nondefense programs.
Several Senate Democrats, including John Tester of Montana, said they were buoyed by the deal but wanted to study the details before signing on.
His House of Representatives equivalent, Nancy Pelosi, told the stories of immigrants in a speech lasting eight hours on Wednesday - setting what is thought to be a new record for the longest speech ever made in the House of Representatives.
Unfortunately for the Senate leaders, the deal faces opposition in the House.
But Republicans and Democrats said they were making progress on a Budget deal that would set new, higher spending limits for defence and non-defence programmes.
White House legislative affairs director Marc Short said the deal would increase government spending by "just shy" of $300 billion over two years.
"Left-wing United States news outlet ThinkProgress said Senate Democrats had pried a "$131bn increase for non-defense programmes" out of their Republican counterparts "in exchange for leaving out a key programme protecting about 800,000 undocumented immigrants". Majority Leader Mitch McConnell, the chamber's top Republican, called the agreement a "significant bipartisan step forward" and suggested that it could help "make 2018 a year of significant achievement for Congress".
The rare bipartisan deal reached by congressional leaders on Wednesday would stave off a government shutdown before a Thursday night deadline and extend the federal government's debt ceiling until March 2019. The first, conservative Republicans, worry the increased spending is fiscally irresponsible.
Rep. Mo Brooks, a Freedom Caucus member, told reporters he is a "hell no" on the Senate deal.
Senate Democrats chose to delay their fight to protect the almost 700,000 young immigrants brought to the country illegally as children, which in part led to last month's three-day shutdown. A budget agreement would nearly certainly mean that many Democrats would support it, thereby guaranteeing that there wouldn't be another shutdown.
"Growing the size of government by 13 percent is not what the voters sent us here to do", the caucus said in a tweet.