The government's funding authority would lapse for the second time in two months if Congress cannot pass a spending bill by the end of Thursday.
Democrats had tied efforts to reach an agreement on spending caps to a deal to protect hundreds of thousands of young immigrants from deportation. Trump wants multiple concessions, including restrictions on legal immigration that Democrats do not support, as part of an immigration bill.
McConnell said Tuesday that the Senate will move forward with debating an immigration plan once leaders can reach a deal to keep the government open.
On Tuesday, Schumer told reporters that he and McConnell made progress toward a spending deal that would raise caps for both defense spending and nondefense programs Democrats have supported. Asked about Trump's shutdown comments, Schumer said he did not want to see government funding lapse again.
"We had one Trump shutdown. Nobody wants another, maybe except him," the New York Democrat said.
After Trump made his comments in the immigration meeting, a member of his party in attendance rebuffed him.
"We don't need a government shutdown on this," said Rep. Barbara Comstock, R-Va. "I think both sides learned that a government shutdown was bad."
Trump responded that, "You can say what you want, we're not getting support from the Democrats on this legislation."
Later Tuesday, White House press secretary Sarah Huckabee Sanders said the White House is "not advocating for a shutdown" and argued that a shutdown would fall on Democrats for "not being willing to do their jobs." Trump wants a long-term spending deal and an agreement on immigration, she said.
She added that she does not expect a spending bill this week to include specifics on an immigration deal.
