(CNN) The White House is dismissing an immigration deal brokered by a bipartisan group of lawmakers as a non-starter just hours before it is expected to be formally introduced in the Senate.

Arizona Republican Sen. John McCain and Delaware Democratic Sen. Chris Coons are slated to introduce a bill Monday that would grant eventual citizenship to young undocumented immigrants who have been in the country since 2013 and came to the US as children, but it does not address all of the President's stated immigration priorities, like ending family-based immigration categories -- which Republicans call "chain migration" — or ending the diversity visa program.

It also would not immediately authorize the $30 billion that Trump is seeking to build the border wall, instead greenlighting a study of border security needs. The bill would also seek to address the number of undocumented immigrants staying in the US by increasing the number of resources for the immigration courts, where cases can take years to finish.

The bill is a companion to a piece of House legislation that has 54 co-sponsors split evenly by party.

A White House official rebuffed the effort, telling CNN that it takes "a lot of effort" to write up a bill worse than the Graham-Durbin immigration bill, but somehow "this one is worse."

Trump tweeted about the latest immigration efforts Monday, writing, "Any deal on DACA that does not include STRONG border security and the desperately needed WALL is a total waste of time. March 5th is rapidly approaching and the Dems seem not to care about DACA. Make a deal!"

Any deal on DACA that does not include STRONG border security and the desperately needed WALL is a total waste of time. March 5th is rapidly approaching and the Dems seem not to care about DACA. Make a deal! — Donald J. Trump (@realDonaldTrump) February 5, 2018

But Coons defended the bill in a conference call with reporters on Monday, calling it a "strong starting place" and a "fresh start" if other talks about immigration don't result in a compromise.

The White House has been aware of the legislation introduced by Texas Republican Rep. Will Hurd and California Democrat Rep. Pete Aguilar for weeks, with officials being informed while it was being drafted and with chief of staff John Kelly and legislative director Marc Short being briefed on the bill in meetings with members of Congress, including Hurd and Aguilar. The Congressional Hispanic Caucus, of which Aguilar is a key member, has been especially supportive of the bill as a compromise, as have moderate Republicans.

But the President has not embraced the proposal, largely for what it leaves out. Hurd and Aguilar, who first described their bill to CNN , have said they intentionally did not seek to appropriate specific funds in their proposal, as neither are on appropriations committees. They have called their bill a "foundation" for conversations about a big deal.

For the border, the bill would create a "smart wall" where the Department of Homeland Security would gain "operational control" of the border by the end of 2020 through "technology, physical barriers, levees, tools and other devices." On family-based migration, the bill doesn't make explicit reference to sponsoring relatives, but the bill authors say that existing law would prohibit parents of these individuals who came to the US illegally to apply for a visa to come back without returning to their home country for at least 10 years before applying and the bill does nothing to erase that requirement.

Coons said that McCain approached him about being a co-sponsor, saying that McCain is deeply concerned about the lack of future certainty for the military because of a budget impasse and the lack of a broader deal on immigration issues, and wanted to find a partner to introduce the Hurd-Aguilar bill in the Senate as part of that effort.

The Delaware Democrat said he recognizes that the bill does not appropriate any money for the border security piece and he'd be willing to look at doing that as well -- and he said he's still committed to the bipartisan Senate talks and is hopeful those could have a breakthrough and a base bill he'd support by the end of the week.

"I remain hopeful that that group can produce a bipartisan deal that is broader than what the McCain-Coons bill is this morning, but in the very real possibility that that does not come together, I think the (bill) is a good base bill," Coons said. "I view the McCain-Coons proposal as a reasonable base bill that would get done the two things we need to get done: ... the status of the Dreamers and border security."

Coons also had some harsh words for the President, saying that despite the White House saying his proposal is the only one that can move forward, "I'm sorry, that's not how the Senate works." He rejected White House criticism of his proposal and said the "worst" thing would be failing to act or only doing a one-year stopgap.

"The President prides himself on being the great dealmaker," Coons said. "Sometimes he makes the greatest contribution when he makes his position known and steps back ... he is least constructive when he does what he did a few weeks ago."

McCain said in a statement the new bill has "broad support."

The Senate is expected to turn to a floor debate on immigration soon, and Coons has been part of a group of bipartisan lawmakers that have been meeting for weeks to try to find a compromise that could pass the vote with more than the 60 votes needed to advance legislation, which would require members of both parties. McCain has been recuperating from cancer treatments but is a veteran of efforts to pass immigration reform.

"While reaching a deal cannot come soon enough for America's service members, the current political reality demands bipartisan cooperation to address the impending expiration of the DACA program and secure the southern border," McCain said in the statement.