S&P 500 futures were up 1%.
The details of the package are under negotiation, but Republicans who control the Senate have proposed sending out checks of as much as $1,200 per-person and $500 for each child, depending on income.
Dow futures (YM=F): 20,996.00, up 388.00 points or 0.8%. Nasdaq-100 futures were up 1.5%.
Democratic Senate Minority Leader Chuck Schumer said, "This is not a moment of celebration but one of necessity".
Any action by the U.S. government would follow aggressive efforts by the Federal Reserve, including its pledge to buy as much government debt as needed to soothe markets, while also lending directly to businesses. In Tuesday's regular session, stocks staged a monster rally, with the battered Dow Jones Industrial Average posting its best day since 1933 by rallying over 11% to close up by over 2,100 points.
Trump emphasized on Tuesday that he was determined to reopen the coronavirus-stricken economy as soon as possible, floating Easter Sunday - less than a month away - as a goal date. I struggle to find out why the second half of the year should be good... This was a machine-driven rally, just like the sell-offs ...
"They want to see the government throw out all the backstops to help support low income or service oriented consumers", said Andrew Smith, chief investment officer at Delos Capital Advisors in Dallas, Texas.
Last week, the Cboe Volatility Index (VIX), also known as Wall Street's fear gauge, eclipsed its financial crisis high and closed at 82.69.
The Australian dollar gained almost 1% during the day to be at 59.10 USA cents and was trading around that level in early Asian dealings on Wednesday.
US stocks opened lower on Monday as a brief spurt of optimism from an aggressive credit boost by the Federal Reserve was overshadowed by the still rapidly spreading coronavirus pandemic. More than 400,000 cases have been confirmed worldwide, including over 50,000 in the US, according to Johns Hopkins University.
Some investors believed the stock market was overdue for a big bounce, having priced in a worst-case scenario regarding the economic damage being done by coronavirus-related shutdowns.
-CNBC's Jesse Pound contributed reporting.