"I want to come up with an agreement and I think my colleagues want to come up with an agreement. Our Republican colleagues know that because they control the House, the Senate and the White House that a shutdown would fall on their shoulders," Schumer told reporters Tuesday.

Summing up the negotiations as "so far, so good," he added that there was "quiet agreement" among congressional leadership that lawmakers should take the lead on reaching a deal.

"What I've said, and I think there's agreement ... if the president doesn't interfere and insist on poison pill amendments to be shoved down the throat of the Congress, we can come up with agreement," Schumer said. Lawmakers are in the middle of a two-week recess. Once they return to Washington on Monday, they will have days to clear a government funding bill before the April 28 deadline.

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A handful of hurdles remain in the government funding talks, including an eleventh-hour fight over key ObamaCare payments.

Democrats are demanding that ObamaCare insurer reimbursements be included in the spending bill, after Trump floated canceling them to try to force a negotiation on healthcare reform.

Schumer told reporters on Tuesday that Democrats are "working hard" on getting the provision included in the bill and are "very hopeful."

"Negotiations seem to be going quite well, and I'm very hopeful we can come to an agreement that everyone can be proud of. I'm not going to get into details, but we're working hard to get that provision in," he said.

Mick Mulvaney, Trump's budget chief, has warned that some of the president's priorities — including defense and border security — should also be included in the bill if lawmakers expect Trump to sign it.

Asked about Trump's request for border funding and immigration enforcement money, Schumer said Tuesday that the "immigration issue should be discussed in the 2018, where there's votes."