Pelosi fired back at Trump on Twitter with a reminder that "800,000 Americans are going without pay".
"Re-open the government, let workers get their [pay] and then we can discuss how we can come together to protect the border," she said.
The squabbling came as the longest government shutdown in history entered its 30th day with prospects for a quick resolution still slim, troubling news for the 800,000 federal employees who have gone without pay and are resorting to food banks, charity and nongovernment jobs to get by.
Yesterday, Trump had offered Democrats three years of deportation protections for some immigrants in exchange for US$5.7 billion in border wall funding. The proposal was immediately rejected by Democrats and derided by conservatives as amnesty.
Nonetheless, Senate Majority Leader Mitch McConnell has said he will move ahead this week on Trump's proposal. He faces an uphill climb in breaking the Senate's 60-vote threshold of a filibuster, with Democrats insisting that they will not negotiate on immigration until Trump reopens the government.
Trump sought today to rebut conservative critiques of his latest proposal, maintaining in a tweet that "No, Amnesty is not a part of my offer."
"It is a 3 year extension of DACA," Trump said, referring to the Deferred Action for Childhood Arrivals programme. "Amnesty will be used only on a much bigger deal, whether on immigration or something else. Likewise there will be no big push to remove the 11,000,000 plus people who are here illegally - but be careful Nancy!"
Trump's reference to amnesty in the tweet could create confusion and is unlikely to help the president get his plan through Congress.
Asked about Trump's tweet, Senator James Lankford, R, said on ABC that he wasn't certain what the President meant. "What I don't know is what the President's talking about there, to say amnesty really involves a much larger group," Lankford said. "That's a longer debate and obviously not something we can solve quickly."
In an interview on Fox News," Vice-President Mike Pence reiterated that Trump's proposal "is not amnesty" because "there's no pathway to citizenship."
"There's no permanent status here at all, which is what amnesty contemplates," Pence said. But in a separate interview on CBS, Pence declined to elaborate on whether Trump's tweet meant that he was leaving the door open to amnesty in the future.
"I'll let the President's words stand," Pence said. He added that Trump is "absolutely determined" to build "234 miles of additional steel barrier" along the US-Mexico border, which the White House outlined in a letter to lawmakers earlier this month.
Other Democrats rallied behind Pelosi and dismissed Trump's latest proposal.
Asked by NBC's Chuck Todd whether Trump's offer signals "progress," Senator Mark Warner said the government must first reopen before any work on border security can commence.
"We cannot reward the kind of behaviour of hostage taking," Warner said. "If the President can arbitrarily shut down the government, he will do it time and time again."
Senator Kirsten Gillibrand, who is pursuing a 2020 presidential bid, said on CNN that her counter-offer to Trump would be "what we put on the table a year ago and voted for, which was to protect all dreamers".
Gillibrand held some similar positions to Trump on immigration in 2008, when she said she was a "firm opponent" of giving amnesty to undocumented immigrants and called for English to be the official language of the United States. CNN's Jake Tapper pressed her on whether those positions - which she now calls Trump "racist" for holding - meant she was racist at the time.
"They certainly weren't empathetic, and they weren't kind, and I didn't think about suffering in people's lives," Gillibrand said. "I realised that things I had said were wrong. I was not caring about others. I was not fighting for other people's kids the same way I was fighting for my own."
In a White House address, Trump had proposed a reprieve on his attempts to end the DACA programme and temporary protected status (TPS) for immigrants from some Latin American and African nations, in exchange for building hundreds of kilometres of barriers on the southern US border and hiring thousands of new law enforcement agents to be deployed there.
Mitch McConnell announced that he would bring the legislation to the Senate floor for a vote in the coming week. But Pelosi dismissed the proposal as a "non-starter" and vowed that Democrats would pass their own legislation to reopen the government, putting the onus on the Senate to follow suit.
