The deal needs approval from both legislative bodies by midnight in order to stave off the second government shutdown in less than a month.
The effort got a boost from President Donald Trump on Wednesday evening.
Kentucky Senator Rand Paul said: "I'm against adding over $1 trillion of debt".
Senator Kirsten Gillibrand (D-NY) declared in a tweet that she will oppose "any long-term spending deal that does not protect" Dreamers: "Senators on both sides of the aisle have a moral obligation to fight for Dreamers as hard as we'd fight for our own families", she asserted, adding, "I have made clear, again and again, that I won't vote for any long-term spending deal that does not protect them". And some tea party Republicans shredded the measure as a budget-buster.
The controversy comes as Republican and Senate leaders managed to agree a two-year budget deal which they hope will avoid another government shutdown. On Wednesday night, the conservative House Freedom Caucus announced its opposition to the bill, saying that "growing the size of government by 13 percent is not what the voters sent us here to do".
But while the bill is expected to pass the Senate, it faces major obstacles in the House where conservatives are appalled at the thought of increasing the federal deficit when the government already is expected to add almost $10 trillion of publicly held debt during the next decade.
The budget deal, which was proposed by Senate leaders on Wednesday, will require Democratic support to pass the House, so the House Democratic leader's comments make the plan's future unclear.
But House Minority Leader Nancy Pelosi said she and many fellow Democrats will oppose the deal unless Republicans allow the chamber to vote on legislation protecting immigrants.
"Very pleased to announce alongside @SenateMajLdr that we have reached a two-year budget deal that will benefit our country in so many ways", Schumer wrote on Twitter, referring to McConnell.
The spending caps would increase about $300 billion over two years, according to a congressional aide.
More funding for disaster relief: The plan includes $90 billion more to be spent on disaster aid for recent hurricanes and wildfires, which have spurred calls from some lawmakers for a more robust federal response.
But Sen. Rand Paul is holding up a vote on the Senate budget deal, accusing Republicans and Democrats of "spending us into oblivion".
As a massive bipartisan budget deal moved towards a vote Thursday, temperatures were rising on the left, where Democrats were fuming that - once again - immigration was being left behind.
The budget agreement would give both the Pentagon and domestic agencies relief from a budget freeze that legislators say threatens military readiness and training as well as domestic priorities such as combating opioid abuse and repairing the troubled healthcare system for veterans.
The core of the agreement would shatter tight "caps" on defense and domestic programs funded by Congress each year. "This budget agreement will give our men and women in uniform the resources they need". Non-defence programmes would receive about 60 billion dollars (£43 billion) over current levels. Those figures would be slightly increased for the 2019 budget year beginning October 1.