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(CNN) —  

A bipartisan group of senators on Wednesday formally unveiled their immigration proposal with new supporters – even as the White House continued to call the deal dead on arrival.

Sens. Dick Durbin, D-Illinois, and Lindsey Graham, R-South Carolina, introduced their bill Wednesday afternoon with Sens. Bob Menendez, D-New Jersey, Michael Bennet, D-Colorado, Jeff Flake, R-Arizona, Cory Gardner, R-Colorado, Lisa Murkowski, R-Alaska, Lamar Alexander, R-Tennessee, Susan Collins, R-Maine, and Mike Rounds, R-South Dakota, putting the bill close to having enough votes to pass the Senate, assuming unanimous Democratic support, but not quite to the 60-vote threshold needed to advance legislation.

The bill appeared to be the same that was presented to President Donald Trump last week, when the President, using vulgar terms, rejected the pitch, according to sources familiar with that meeting.

If passed, the bill would appropriate $2.705 billion in border security improvements, eliminate the visa lottery, make permanent the Deferred Action for Childhood Arrivals program – offering a pathway to citizenship to those who qualify – and limit “chain migration,” or family-based migration, of the individuals eligible for the program.

“We’d be crazy to want them to leave. If you met any of these Dream Act kids, the last thing you’d want is for them to leave,” Graham said on the Senate floor Wednesday evening. “At the end of the day there’s a deal to be had; it just needs to be done. The reason this bipartisan group that I’m part of came about is because no one was doing much of anything.”

But the President himself and his top advisers continued to tell reporters on Wednesday that the bill was not what the White House was looking for, virtually assuring the measure had no chance of moving forward as it is.

Senate Majority Leader Mitch McConnell on Wednesday reiterated that without an affirmative endorsement from Trump, a bill would not move for a vote.

“Well, at the risk of being repetitious, I’m looking for something that President Trump supports,” McConnell told reporters. “And he’s not yet indicated what measure he’s willing to sign. As soon as we figure out what he is for, then I would be convinced that we were not just spinning our wheels going to this issue on the floor, but actually dealing with a bill that has a chance to become law and therefore solve the problem.”

White House chief of staff John Kelly said Wednesday that the bill didn’t meet the bipartisan and bicameral standard set by the President.

“His sense was, two things. It fell short of certainly what he was looking for based on the Thursday conversation and other conversations,” Kelly said. “But more to the point, it did not include all of the senators that have been involved in all of the discussions about DACA and certainly did not involve the House.”

Sens. Thom Tillis, R-North Carolina, and James Lankford, R-Oklahoma, said in a joint statement Wednesday evening they would oppose the bill because it “falls short” on border security reforms.

“Unfortunately, the ‘Gang of Six’ proposal falls short since it fails to include even basic border security reforms,” they said in a joint statement. “Threatening a government shutdown or demanding a vote on a bill that is doomed to fail is counterproductive. It is also unfair to give false hope to the many DACA recipients waiting on Congress to finally enact legislation.”

Conservative hard-liners Senate Judiciary Chairman Chuck Grassley and Sens. Tom Cotton, R-Arkansas, and David Perdue, R-Georgia, also slammed the bill in a joint statement. They were previously negotiating with the group, but were left out of meetings over disagreements.

“The Durbin-Graham-Flake proposal would do nothing to solve the underlying problem in our current immigration system,” they said in a statement. “It does not take the needed steps to fix our porous border and it fails to empower law enforcement to apprehend and remove dangerous criminals who are here illegally.”

The President said in an interview with Reuters on Wednesday that the bill fell short.

“It’s the opposite of what I campaigned for,” Trump said.

And White House legislative director Marc Short, coming out of lengthy meetings with lawmakers on different sides of the issue, said the President was still looking for a deal that covered only existing DACA recipients, not a broader population, and a curtailing of family-based migration that goes beyond only those affected by a DACA measure – two elements inconsistent with the Durbin bill.

CNN’s Ted Barrett contributed to this report.