"As the front-runner of the Republican nomination, Donald Trump and his proposals are leading the public debate in our country," Reid said from the Senate floor. "Republicans who support these illogical plans should be prepared for the next logical step: Voting on his vision of America."

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The Senate is expected to take a procedural vote Wednesday afternoon on the House legislation. The proposal would effectively freeze the acceptance of Syrian and Iraqi refugees until the administration could certify they aren't a national security threat.

Reid had previously suggested they would try to force a vote on Trump’s Muslim ban, though he had not indicated when or where such a vote would come. Democrats are using McConnell's pledge that Republicans would open up the amendment process to try to pressure him to accept their deal.

Reid added that Democrats "will seek to advance a limited number of amendments on this bill... not tons of amendments, not scores of amendments, but four or five amendments." Democrats could also offer amendments on blocking suspected terrorists from being able to buy a gun and increasing funding for law enforcement antiterrorism efforts.

Republicans will need the support of at least six Democratic senators to overcome Wednesday's hurdle. Democrats could still block the legislation when senators try to end debate.

McConnell pressured Democrats to "resist" any political pressure to object to the legislation, calling it "intensely shortsighted."

"It is certainly worrying to hear that Senate Democrats are now being pressured to block us from even debating it," he added.

Reid also took a shot at Republican Sens. Marco Rubio (Fla.) and Ted Cruz (Texas), who are both running for president, saying it was a "big day" in the Senate because they are both expected to be in the Capitol on Wednesday.

"If for any reason the Republican leader needs more time to discuss the proposal with his caucus, we're happy to reschedule the vote. Now, I know it's a big day in the Senate," he added. "I know we have a tight schedule because they're only going to be here probably for an hour or two."

Pressed by Reid on his proposed deal to move forward on the legislation, McConnell added that "obviously, we'll be talking to the Democratic leader about a way forward on the bill, and we'll have those discussions and report back later."

Reid had suggested last week that Democrats would block the refugee bill. But, a Democratic aide told The Hill last week that Senate Democrats were undecided on if they would block the legislation, suggesting the proposal could be a vehicle for the Trump fight.