The two senators said the plan would provide $25 billion over a decade for a wall and other border security measures, an amount Trump has requested.
Meanwhile, a weeklong congressional recess begins next week, and debate on immigration is only expected to continue for this week, according to Senate Majority Whip John Cornyn, R-Texas, who helps control the floor schedule.
On Tuesday, Democrats blocked Republicans from setting up votes, while GOP senators themselves outlined different approaches for how to build a compromise that could get 60 votes - the supermajority required to overcome a filibuster and clear legislation from the Senate.
"We still have to solidify our caucus", Durbin said. Flake. "Cutting legal immigration in half or by a third is just a nonstarter for me and a lot of Republicans and certainly all the Democrats".
"Instead of making a proposal in good faith or working with Democrats on a compromise, President Trump is trying to force his unpopular, hard-line immigration agenda down the throats of the American people by calling it a DACA bill", Schumer said. "We went as far as we could in that direction, but any more and the House would never take up the bill and the president would not be able to sign it". Charles Grassley and six other Republicans that includes his "four pillars".
The dealmaking came after Democrats spent the first three days of immigration week blocking all floor action and retreating behind closed doors to try to work out another "grand bargain", similar to the deals the Senate tried on immigration in 2007 and 2013.
Ending family-based migration must also be opposed, the letter says, because it would cut overall immigration between 33 percent and 44 percent and "would needlessly penalize American families".
The Grassley bill is unlikely to win support from many Democrats.
President Donald Trump speaks to media during a meeting with bipartisan Members of Congress about infrastructure in the Cabinet Room of the White House in Washington, Wednesday, Feb. 14, 2018.
Their pitch was as unified as it was simple: There is only one proposal the President supports, there is only one proposal the President will sign and as such, there's only one proposal the more conservative House Republican leadership would be willing to take up.
The centrist senators on Wednesday said that they had finished the contours of an agreement that would protect "Dreamers" in exchange for more spending on border security.
By and large, Democrats are looking to provide protections and a path to citizenship for Dreamers, and Republicans are looking to bolster border security and reduce the number of immigrants entering the US through family reunification (in which immigrants can sponsor family members to join them in the U.S.).
He urged senators to oppose any legislation that fails to fulfil all four pillars and to oppose any "short-term Band-Aid approach".
Also in play is a more modest plan by Sens.
"I wouldn't bet my auto, but I would bet lunch that they might not end up with anything by the end of the week, " said Mark Krikorian, executive director of the Center for Immigration Studies, which seeks reductions in immigration.
The Democrats shut down the government for the Dreamers, and Nancy Pelosi staged an eight-hour speech in support of illegals -They are invested in a deal. She was referring to top Trump adviser Miller, who is believed to heavily influence the president with hardline immigration views. The sooner we get started, the better, because we'll need to wrap this up this week.
Another proposed bill closes the door for legal immigration for undocumented parents who brought their children into the country illegally. It said the proposal would "increase illegal immigration, surge chain migration" - a term critics use to describe family-based immigration - and cause other problems.
But it would only make limited changes to family reunification, and leave the diversity lottery untouched because it is too "politically toxic", Graham said. John McCain, R-Ariz., and Chris Coons, D-Del., and supported by Senate Minority Leader Chuck Schumer, D-N.Y.
Rep. Mark Meadows, the North Carolina Republican who leads the Freedom Caucus, said the future of Mr. Ryan's speakership could rest on how he handles the issue. Pat Toomey (R-Pa.) that would crack down on sanctuary cities, and a placeholder for the Collins proposal.