According to media reports, senators will vote to end the shutdown with a short-term spending bill that would last three weeks.
McConnell said Sunday that Congress must stop victimising the American people and get back to work.
Senator Schumer, on the other hand, said that the Democrats and the GOP are "yet to reach an agreement on a path forward."
House and Senate lawmakers met throughout the day Sunday to end the government shutdown as the impasse continued into the second day, with both chambers hoping to strike a deal on spending and immigration that would reopen federal agencies ahead of the work week.
House Speaker Paul Ryan said the lower chamber has agreed to accept a short-term deal that would fund the government through Feb. 8 if the Senate is able to pass such a bill.
Senate Democrats have so far not agreed to support a bill that makes no concessions on immigration.
President Donald Trump has appealed to the Senate to abolish the filibuster. (ANI)