Some Republican senators, including Lindsey Graham of SC, voiced discontent over the expansion of unemployment insurance benefits, arguing that this could discourage people from returning to work.
The motion failed on a 48-48 vote.
Hospitals and health systems across the country are expected to get $130 billion in emergency grants, while billions more would be spent on medical and personal protective equipment.
Later, Trump appeared to retreat from his Easter goal at a news conference alongside Fauci.
Democrats scoffed in response, noting that employees can not collect unemployment if they leave their jobs voluntarily.
"Why would the senators hold up this really important bill. because they resent people at the low end of the spectrum who have lost their jobs, from getting $600?" On the other side of Capitol Hill, House Speaker Nancy Pelosi (D-Calif.) has already endorsed the broad contours of the Senate compromise, but many House members are unavailable, especially after their recent interactions with two lawmakers who've since tested positive for COVID-19. She said leaders pledged to give 24 hours' notice ahead of the vote and said on CNN "we can take a voice vote if someone calls for a recorded vote then we're prepared to go in that direction as well".
Treasury Secretary Steven Mnuchin has been meeting with Senate Minority Leader Chuck Schumer to resolve Democrats' complaints about the Republican legislation.
This is, of course, a far more progressive model than the one pitched by Senate Majority Leader Mitch McConnell (R-Ky.) late last week.
"Help is on the way. Hopefully this will be won quicker", he said.
It was unclear, however, how quickly the bill would get to Trump's desk.
Missing from Wednesday's votes was No. 2 Senate Republican John Thune, who said in a statement he did not feel well when he woke up on Wednesday and chose to take a charter flight home to South Dakota "out of an abundance of caution". The relief bill is valued at $2 trillion and provides relief for the economic fallout from the coronavirus.
NBC News' report added, "The bill is also expected to include roughly $100 billion in assistance for hospitals; $350 billion in assistance to small businesses; $500 billion in aid for corporations, including airline companies and cruise lines, that have been hurt by the outbreak; and about $150 billion for state and local stimulus funds". The money at stake amounts to almost half of the $4.7 trillion the U.S. government spends annually.
Investors were cheered by the news of the deal. On Wall Street, the benchmark S&P 500 rallied for a second straight day, closing up 1.15%. Those whose incomes are higher will see smaller checks. The World Health Organization said the US has the potential to become the new epicenter of the global pandemic as the number of known infections soars.
Former U.S. Federal Reserve Chairman Ben Bernanke told CNBC on Wednesday that the U.S. economy could have a "very sharp" recession in the next two quarters, but followed by a "fairly quick rebound", if there's not too much damage done to the workforce and the businesses during the shutdown.
Pelosi said she was sympathetic to Cuomo and other state officials, but wanted a rescue package to move on.
As of the evening of March 25, there were 65,000 diagnosed cases of the highly-contagious virus in the USA and more than 900 deaths.
But Cuomo said the Senate package would send less than $4 billion to NY, far short of his estimate that the crisis will cost his state up to $15 billion over the next year. The sweeping orders are aimed at slowing the pathogen's spread, but have upended daily life as schools and businesses shutter indefinitely.