The measure failed to clear a procedural vote in the Senate just after 10 p.m. ET Friday.
Graham said he will not support a temporary funding bill passed by the House when the Senate takes it up Friday.
"We had a long and detailed meeting," Schumer told reporters at the Capitol after the meeting.
Earlier Friday, President Donald Trump urged Senate Democrats to get behind a short-term bill and added, "shutdown coming?"
Mick Mulvaney, chief of the Office of Management and Budget, said that odds of a shutdown occurring are 50-50.
The Senate is divided over how to proceed on a government funding bill with only hours until a shutdown.
The bill now heads to a Senate where lawmakers face a tougher task in extending government funding
Ryan's comment came as doubts emerged about the Senate's ability to approve the same short-term funding legislation.
A tweet by President Donald Trump on Thursday morning threw more confusion into the effort to avoid a shutdown.
House Minority Leader Nancy Pelosi takes questions Thursday as Congress barrels toward a government shutdown this weekend.