It goes next to the House.
Trump has not yet said he will support the measure, but the White House says he'll take a look.
"Or, rather I should say, that is the $5 billion question", he added.
Even if DHS were temporarily left without funding, though, it would not materially impact border security, because Border Patrol and ICE agents would nearly certainly still be required to report for work.
Without a resolution, more than 800,000 federal employees could be furloughed or ordered to work without pay. McConnell said a stopgap measure could be approved, though he suggested that House Minority Leader Nancy Pelosi, who is poised to become House speaker when the Democrats take control January 3, would not want to saddle the new year with a budget brawl.
The bill, which was approved by the US Senate late Wednesday, was expected to keep the government funded and open for business through February 8, 2019.
"The president said, 'I will not sign this bill.' So we're going to go back and work on adding border security to this". If that proves impossible, lawmakers hope to extend the program in a separate bill addressing a maintenance backlog at national parks and other public lands issues.
"This is a missed opportunity to pass full-year funding bills now", she said. The government will partially shut down at midnight Friday unless there's a deal.
After a backlash from immigration hawks in Trump's base - as well as House Republicans who support building the wall - the president signaled Thursday that he would not sign a bill that did not provide funding for at least some border security.
Senate Majority Leader Mitch McConnell, R-Ky., said the Senate would remain in session Thursday.
We just had a very long productive meeting with the president.
On Wednesday, McConnell attacked Democrats for rejecting it, saying, "It seems like political spite for the president may be winning out over sensible policy". He calls it "a simple measure" to show that Republicans are governing.
Schumer said Democrats oppose Trump's border demands because the wall is "inefficient" and because Trump, as a candidate, promised that Mexico would pay for it, which Mexico has refused to do.
Radio host Rush Limbaugh said Mr Trump is going to "get less than nothing". Trump appeared to back off his demand for $5 billion to fund a border wall on Tuesday after days of impasse with Congress. The U.S. Postal Service, busy delivering packages for the holiday season, wouldn't be affected by any government shutdown because it's an independent agency.
Several GOP senators said that even if the House passed additional funding for border security, it could not pass the Senate, where votes are needed from Democrats to advance it.
Press secretary Sarah Huckabee Sanders says that "at this moment, the president does not want to go further" on a proposed budget deal "without border security, which includes steel slats or a wall".
So, after being called out by his own personal YouTube network, Trump let House Republicans know that he wouldn't be signing anything because Fox News runs the White House and he's a Russian bot that can only do what he's told.
However, the standoff between Trump and Congress on the border wall continues.