Conservatives are wary, fearing he will strike a soft compromise.
The NAACP is accusing Trump of "lowbrow, callous and unfiltered racism".
"We were hoping for that, but the president is not prepared to do that at this moment", Durbin said.
Cotton said the plan falls short of Trump's request for border wall funds.
President Donald Trump reportedly lashed out in a Thursday meeting with lawmakers about immigration reform, demanding to know why the U.S. should accept citizens from what he called "shithole" countries.
The White House did not deny the account of Trump's use of language, but instead suggested the president was "fighting for permanent solutions" that strengthen the nation, in part through the use of a merit-based immigration system.
Trump reportedly questioned why the USA would want to admit more people from "shithole countries". The Trump administration has proposed $18 billion over a decade to build a wall.
President Donald Trump signaled Tuesday he could support a deal to allow hundreds of thousands of immigrants living in the country illegally to keep their status, but only as part of a broader border security deal.
Some argue we shouldn't read too much into Trump's meeting, saying he was merely engaging in 4-D chess or "Art of the Deal" negotiations. Some of the most visible senators include numerous potential contenders for the Democratic presidential primary in 2020, like Sens. Richard Durbin told Trump that under the proposal, a lottery for visas would be ended.
"How do six people bind the other 94 in the Senate?". As for Africa, he asked why more people from "shithole countries" should be allowed into the US, the sources said.
Trump originally said Mexico would foot the bill for the wall, a claim incredulously dismissed by the U.S.'s southern neighbor.
Trump's remarks were remarkable even by the standards of a president who has been accused by his foes of racist attitudes and has routinely smashed through public decorum that his modern predecessors have generally embraced.
If Trump wants a big immigration deal now, he'll have to "give a lot more to both Democrats and Republicans who want a more compassionate approach", said Ali Noorani, executive director of the advocacy group National Immigration Forum. "We are now working to build support for that deal in Congress".
The significance of their agreement was initially unclear.
White House spokeswoman Sarah Sanders said, "We still believe we can get there".
Graham and Durbin are leading efforts to codify protections for so-called "dreamers", immigrants who came to the United States illegally as children.
President Trump on Thursday organized a meeting to hear the plan, which includes $2.7 billion for border security (including the long-promised wall). But there's also talk from Republicans, including the president, of attaching conditions to any deal - like expanding the border wall with Mexico. Cornyn heads up an all-Republican working group that includes Sens. "Now, absent an injunction, they will slide back to the pre-DACA era and associated hardship".
But Rep. Ruben Gallego, D-Ariz., who was not there and like Cuellar is a member of the Congressional Hispanic Caucus, said he favored a narrow bill protecting Dreamers with perhaps something negotiable on border security.
Spokesman Jason Samuels says, "Sen". And let's see if we can get something done.
The next step is seeking White House reaction.
A senior Republican congressional aide, who did not want to be named, noted that the White House was briefed on the senators' proposal and said, "It's clear it's a non-starter".
Pelosi's talking about a group of lawmakers that's blessed by President Donald Trump. "Brought to this country as children by undocumented parents or family members, they are not "illegal" through any fault of their own", Archbishop Gomez wrote.
There're about 50,000 of those that are allowed to bring in people from countries that usually don't come into the United States, so we made a decision to narrow it down into phase one, which are those four items.
Pelosi asks, "Why now?"
Trump also offered a clarification later in Tuesday's meeting - "We do a Phase 1, which is DACA and security, and we do Phase 2, which is comprehensive immigration" - a relieved-looking McCarthy all but leaped from his seat, pointing at Trump like a teacher whose promising student, after several false starts, finally has alighted on the correct answer. They're under pressure to find a breakthrough before a deadline next week that could lead to a government shutdown. In exchange, Trump and Republicans want toughened border protections and tightened restrictions on others trying to migrate to this country.
Four dead, dozens missing after Venezuelan migrant boat sinks
Around 30 people were on the boat when it sailed from La Vela de Coro in Venezuela's northwestern Falcon state, sources said. Regional Civil Protection director Jose Montano said the boat departed Venezuela for the Dutch Caribbean island on Tuesday.
Friend of U Penn student arrested in murder case
They met and drove to the Hobby Lobby parking lot in Lake Forest to "hang out" and "catch up", and then went to the park. According to the document, Woodward said Bernstein seemed depressed, but didn't say he wanted to hurt himself.