But Trump tweeted, "No, Amnesty is not a part of my offer".
The criticism from both sides complicates Trump's effort to win at least some Democratic support.
NBC's Geoff Bennett argued that these proposals would be unlikely to change the stance of Democratic lawmakers, who have supported Obama-era immigration programmes for years.
"I think it's totally impossible", Elmendorf said, when asked if he saw any chance of seven Senate Democrats backing Trump's offer.
The proposal would also extend temporary protections for some young people brought to the US illegally as children and immigrants who fled disaster zones. But Trump, who has yet to acknowledge that offer, laid out his own plan, which officials said had been in the works for days.
The Trump administration has taken certain steps to blunt the impact of the shutdown, effects of which will nevertheless multiply over time, including at airports where Transportation Security Administration employees are increasingly calling out sick.
Some airports experienced longer wait times at security checkpoints, and Baltimore/Washington International Thurgood Marshall Airport closed one of its checkpoints because of excessive absences.
Still, TSA reports that, on Saturday, 99.9 percent of passengers waited less than 30 minutes and 93.8 percent waited less than 15 minutes.
Similar objections a year ago torpedoed an even bigger deal that would have created a path to citizenship for millions of undocumented immigrants while also fully funding Trump's border wall. About 12 million people are living in the United States illegally, according to U.S. Department of Homeland Security estimates.
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Trump's son-in-law and senior aide, Jared Kushner, has led the work on the proposals, said three people familiar with White House thinking who were not authorized to speak publicly. President Trump even warned Pelosi that her inaction could force the president to take action against illegal immigration by upping deportations. "Amnesty will be used only on a much bigger deal, whether on immigration or something else", Trump said on Twitter.
US President Donald Trump is set to propose a deal meant to end a 29-day partial government shutdown, a source familiar with his plan says. He claimed that Pelosi had made up her mind before she even heard the details. Leaders insisted Trump needs to open the government before beginning any negotiations over border security or barrier funding.
"I don't take him on his word on anything", Ms. Gillibrand said.
On Jan. 19, Trump outlined his new offer, which sought to provide temporary protection from deportation to some 700,000 illegal immigrants in exchange for $5.7 billion in border wall funding.
Beyond negotiating with Democrats, the president also noted that he has already built sections of the wall without Congressional funding and he could continue to do so.
Pelosi and other Democrats have rejected Trump's latest proposal and remain opposed to the idea of building a wall along the southern border with Mexico. Workers have now missed one paycheck and will miss another if the shutdown is not resolved in the next week.
Republicans have a 53-seat majority in the upper chamber and need seven Democrats to vote yes in order to move the president's deal on the path to passage. Mitch McConnell pushes it. Trump's proposal also includes a three-year extension of the Temporary Protected Status (TPS) that now lets people from certain countries stay in the United States if they left their homes to escape wars, disasters, or "other extraordinary and temporary conditions".
The legislative process is a negotiation and up to this point, literally for the last month while the president and I have stayed here in Washington and been engaging continuously with Democrat leadership, and with rank-and-file members in the House and Senate. Lisa Murkowski, R-Alaska, said she would study the details of the plan but did not commit to vote for it.
Trump is under pressure to end the shutdown, with Americans increasingly blaming him for refusing to sign spending bills that would provide pay for 800,000 federal workers who have been idled or working without pay for almost a month.
Temporary protected status is given to nationals from designated countries affected by armed conflict or natural disaster.
Vice-President Mike Pence, who has been leading the administration's contacts with Congress, said the Senate would put the proposal to a vote as early as Tuesday.