Washington (CNN) The Senate will hold a vote on immigration legislation, including whether to extend the Deferred Action for Childhood Arrivals program, next month, but only if lawmakers can reach a deal before the end of January, Majority Leader Mitch McConnell said Wednesday.

"There are bipartisan discussions in the Senate, involving the administration, about improving border security, interior enforcement and reforming important parts of our broken immigration system, including addressing the unlawfully established Deferred Action for Childhood Arrivals program," McConnell said in a statement.

"If negotiators reach an agreement on these matters by the end of January, I will bring it to the Senate floor for a free-standing vote," he said.

McConnell's statement came after Arizona Republican Sen. Jeff Flake had said he received a commitment for a vote on immigration. Flake's statement and tweet came after he voted for the GOP tax bill that passed early Wednesday morning.

Flake had been hesitant on the tax package, but in negotiations said one of his requirements had been a commitment to progress on a plan to provide a permanent legislative version of DACA, which President Donald Trump is ending.

Read More