The "Dreamers" are young people who were brought to the United States illegally as children by their parents or other adults, mainly from Mexico and Central America, and who mostly grew up in the United States.
Trump said last year he would end DACA on March 5 and asked Congress to come up with a legislative fix before then that would prevent the Dreamers from being deported.
Democrats have withheld support for a temporary funding bill to keep the government open over the DACA issue. McConnell extended them an olive branch on Sunday, pledging to bring immigration legislation up for debate after Feb. 8 so long as the government remained open.
Senate Democratic leader Chuck Schumer objected to that plan and it was unclear whether McConnell's pledge would be enough for Democrats to support a stopgap funding bill.
Congress failed last year to pass a complete budget by Oct. 1, the beginning of the federal fiscal year, and the government had since been operating on a series of three stopgap spending bills.
Republicans control both the House of Representatives and the Senate, where they have a slim 51-49 majority. But most legislation requires 60 Senate votes to pass, giving Democrats leverage in that chamber.
Trump earlier this month told a bipartisan Senate working group that he would sign whatever DACA legislation was brought to him. The Republican president then rejected a bipartisan measure and negotiations stalled.
McConnell had previously insisted that the Senate would not move to immigration legislation until it was clear what could earn Trump's support.
Republican Senator Jeff Flake, who is involved in bipartisan immigration negotiations, said McConnell's statements on Sunday indicated progress in negotiations and he urged his Democratic colleagues to approve another stopgap bill.