"It is clear that Congress has at least until March -- at a minimum, and maybe even longer -- to reach a compromise on the DACA question," McConnell said, citing the court order that froze the wind-down of the program and led to the administration taking new renewal applications from the young undocumented immigrants it covers.
"There is no reason my colleagues should hold government funding hostage over the issue of illegal immigration," McConnell said.
If Congress doesn't pass a funding bill by Friday, the government could be heading for a shutdown on the one-year anniversary of Donald Trump's presidency.
Democrats have signaled they wouldn't be quick to fund the government without a guarantee that there is a plan on the way to protect hundreds of thousands of young recipients of DACA, which expires in March.
After Trump's incendiary language
and his rejection of a bipartisan deal last week, Democrats may be leaning more toward withholding their votes on a spending compromise.
But at least one Democrat said he'll vote to support a continuing resolution. Sen. Joe Manchin, a red state Democrat up in 2018, told reporters Tuesday he will vote to keep the government funded even without a DACA deal.
"I'll vote to keep the government open," the West Virginia senator said.
If the House passes a stopgap measure to keep the government running ahead of the Friday deadline, McConnell will need at least nine Democrats in the chamber to vote "yes" again on a short-term funding bill -- and he may need more, depending on whether any Republicans vote against the continuing resolution or are absent due to health reasons.
Democratic Sen. Cory Booker of New Jersey said he won't support a short term spending bill that doesn't include DACA fix.
"I will vote against a clean (continuing resolution) if it does not include DACA, a fix to some of the other things that are high priorities to me. Why are we kicking the can down the road?" Booker said.
Other Democrats who supported the last stop gap measure in December said Tuesday they were unsure where they stood on plans to extend funding again.
Sen. Angus King, an independent from Maine who caucuses with Demcorats, told CNN that he is reluctant to vote for another CR if it doesn't include DACA, but that he has not made up his mind yet.
"I'm just tired of voting for CRs," King said.
He added that he also wants to see re-authorization of the Children's Health Insurance Program to be included, which Republicans could use as a bargaining chip to attract red state Democrats.
Sen. Tim Kaine of Virginia wouldn't say Tuesday night whether he'd support the short term bill without a DACA fix.
"We want to see what's in it and then we're going to decide," he said. "We need to do a DACA deal. The bipartisan deal that was announced was a good one. We support it."
This story has been updated and will continue to update with additional developments.