Another official also said that despite suggestions from Senate Democrats, the White House's framework is "galaxies apart" from what Senate Minority Leader Chuck Schumer proposed to the President over the weekend.
Officials said they expected McConnell to bring the president's plan to the Senate floor for a vote during the week of February 5, just days before the February 8 expiration of a short-term government spending plan.
The White House said earlier on Wednesday that it would unveil a framework for immigration legislation that it believes can pass muster with both Republicans and Democrats in the U.S. Congress, hoping to resolve the long stand-off over the fate of Dreamers that contributed to the brief government shutdown this week.
Trump, whose tough immigration stance was a key part of his 2016 presidential campaign, said in September he was ending the Deferred Action for Childhood Arrivals, or DACA, program for Dreamers that was created by Democratic predecessor Barack Obama.
The plan would also end the diversity visa lottery program.
Evangelicals leaders are celebrating Donald Trump's announcement of a "pathway" that could allow hundreds of thousands of people, brought into the United States illegally as children, to apply for citizenship. "I just don't think they should be used as bargaining chips". We're going to solve the problem'.
It was a reversal for the president, who had previously said he was opposed to a pathway to citizenship for Dreamer immigrants.
Activists react to potential changes for dreamers
Senior White House officials who briefed reporters Thursday on the framework also expressed a pointed rejection of the Durbin-Graham bill that the White House rejected in recent weeks.
And the outcry from both sides doesn't bode well for the prospect of Congress reaching a deal to address DACA before it expires March 5 - the arbitrary deadline Trump put on the program previous year. Trump tweeted. Critics said the Trump framework was not helpful to the bipartisan Senate negotiations on an immigration deal. Many of them should have $11,111 on hand.
Other Republicans said citizenship for Dreamers could not be discussed without considering other aspects, including border security. "I truly appreciate President making it clear that he supports a path to citizenship for DACA recipients".
"DACA has been made increasingly hard by the fact that Cryin' Chuck Schumer took such a beating over the shutdown that he is unable to act on immigration!"
"I get scared and anxious but I have had nothing but support through this", she said.
Cruz did not mention Trump by name, but there is no doubt that he was voicing the concern shared by numerous men and women who supported Trump in the last presidential election.
That could make the plan hard to get past Democrats, no matter how strong they want the Dreamers reform. It expands the amnesty from the 700,000 beneficiaries of DACA to a larger pool of 2 million, in exchange for more resources and legal authorities at the border and changes in the legal-immigration system.