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"Should these issues not be resolved by the time the funding bill before us expires on Feb. 8, 2018 , assuming the government remains open, it would be my intention to proceed to legislation that would address DACA, border security and related issues," McConnell said on Sunday night, referring to the Deferred Action for Childhood Arrivals program.

McConnell noted that he hopes to get the immigration fight, as well as separate discussions on disaster relief, health care and the budget caps, finished "as quickly as possible."

McConnell's announcement comes as a bipartisan group of senators met on Sunday to discuss a possible path to getting 60 votes for a government funding bill. The bill would include the GOP leader making a commitment to move to immigration if they weren't able to get a larger deal by Feb. 8 .