He said that if an agreement isn't reached on a myriad of issues by the next funding deadline, Feb. 8, then it is his "intention" to bring up the legislation.
"The shutdown should stop," McConnell said, saying he wanted to "move on to other business that's important to our country."
"However, should these issues not be resolved by the time the funding bill before us expires on Feb. 8th, 2018, assuming that the government remains open, it would be my intention to proceed to legislation that would address DACA, border security and related issues," he said. The other issues include defense spending and disaster relief.
McConnell proposed a 10 p.m. vote Sunday, instead of the originally scheduled for 1 a.m. Monday, but Senate Democratic leader Chuck Schumer of New York objected, saying, "We have yet to reach an agreement on a path forward that would be acceptable for both sides."
McConnell's statement was enough for two Republicans, Lindsey Graham of South Carolina and Jeff Flake of Arizona, to say they would support the bill. They were two of the four Republicans who helped to block the short-term funding bill on Friday, which led to the shutdown.
Democrats set a meeting Monday morning to hear from Schumer on the specifics of what is being proposed. Many Democrats said they didn't trust McConnell to follow through on his "intention" to allow a vote on DACA, pointing to McConnell's promise to Sen. Susan Collins, R-Maine, during the tax debate to allow a vote on legislation to stabilize the insurance markets under the Affordable Care Act, which hasn't yet happened.
"He's broken too many of those commitments previously," Sen. Jeff Merkley, D-Ore., said Sunday on MSNBC.
Despite the lack of progress, negotiations late Sunday appeared to have moved slightly toward agreement to reopen the government.
The movement came after a day that started pessimistically, with leaders of both parties blaming one another for the shutdown on the Sunday morning shows, but it grew more optimistic as a bipartisan group of senators hashed out a plan.
The 22 senators, organized by Collins, emerged from their second day of talks and marched their proposal to Schumer and McConnell. Collins was instrumental in ending the last government shutdown, in 2013. While McConnell and then-Democratic leader Harry Reid were stuck in a stalemate, she forged a bipartisan path forward then.
The two leaders huddled in private, on neutral territory behind the Senate chamber, to discuss the proposal. It was the first time they'd done so since the shutdown started at midnight Friday. They didn't leave with a deal, according to Schumer's spokesman, Matt House, but the standoff seemed to have softened slightly.
Related: Tom Cotton: Support for immigration deal won't hinge on Trump
Monday's noon vote would reopen the government by extending funding for three weeks, until Feb. 8. It would also extend the low-income children's health insurance program, CHIP, for six years and suspend some taxes under the Affordable Care Act.
"First prize in government shutdown is you get to be dumb, not dumber. That's the best you can hope for — not to be looked at as the dumbest guy in the room — or gal," Sen. Lindsey Graham, R-S.C., told reporters.
The stark division between the parties was on full display Sunday morning as lawmakers took to the Sunday shows to assign fault for the funding impasse on their political opponents, while offering little optimism that a compromise is on the horizon.
"You can't blame Donald Trump for Senate Democrats' shutting down the government. They shut down the government with no endgame in sight," House Speaker Paul Ryan said on CBS's "Face The Nation."
Related: Government shutdown 2018: What's open and what's closed?
Democrats, meanwhile, have dubbed the stalled negotiations the "Trump shutdown" and say the Republicans' control of the White House and Congress puts the blame squarely on their shoulders.
"I wish the president would help us. At some point, his leadership could make the difference," Sen. Dick Durbin, D-Ill., said on NBC's "Meet The Press." He declined to predict whether the government would be funded by Monday night.
Still, the heart of the deliberations hascentered on immigration reform and the protections for so-called Dreamers, the children of undocumented immigrants. Trump moved to undo the Obama-era program, which is set to expire March 5, and the White House has said any move to reauthorize it needs to be accompanied by funding for a wall between the United States and Mexico.
"This shutdown is going to get a lot worse tomorrow," McConnell said in a speech on the Senate floor. "Today would be a good day to end it."
Schumer called on Trump to return to the bargaining table. "I'm willing to seal the deal, to sit and work right now, with the president or anyone he designates. Let's get it done."
Schumer said: "This is the Trump shutdown. Only President Trump can end it. We Democrats are at the table, ready to negotiate. The president needs to pull up a chair and end this shutdown."
But McConnell was quick to note that it's Democrats in the Senate who are using procedural rules to block passage of the short-term bill and Republicans are calling the standoff the "Schumer shutdown."
Skeptical Democrats maintain that the weeks-long spending bill being considered in the Senate is just a stalling tactic that will not lead to a serious legislative debate about immigration reform.
"We need to have clear assurance that if we can pass ... a bill in the Senate, it'll be taken up in the House and won't be ignored," Durbin said Sunday.
Vice President Mike Pence, however, signaled in a speech to U.S. troops stationed in the Middle East that immigration reform isn't on the table until the government is up and running again.
"We're going to demand that they reopen the government," Pence said. "In fact, we're not going to reopen negotiations on illegal immigration until they reopen the government and give you, our soldiers and your families, the benefits and wages you've earned."
CORRECTION (Jan. 21, 2018, 11:24 p.m. ET): A headline in an earlier version of this article misstated the length of the government shutdown. It has lasted nearly two full days, not three days.