Japan's benchmark fell nearly 3% and other regional markets were mostly lower. Shares in Australia surged 7% after the government promised more recession-fighting stimulus.
"We want to keep the engine of our economy running through this crisis," Prime Minister Scott Morrison told reporters in Canberra.
The unprecedented $130 billion package includes wage subsidies of up to $1,500 per two weeks to businesses to keep workers on the job.
U.S. futures rebounded, gaining nearly 1%, but oil prices were lower.
Tokyo's Nikkei 225 lost 2.8% to 18,851.04, while the Kospi in South Korea reversed early losses, gaining 0.4% to 1,717.10. The Shanghai Composite shed 0.7% to 2,752.53, while the Hang Seng in Hong Kong lost 0.4% to 23,399.38.
Shares fell in Taiwan and Southeast Asia. India's Sensex fell 2%.
Hopes that a $2 trillion relief bill would ease the economic havoc brought by the pandemic fell flat on Friday, as the major indexes ended lower. The S&P 500 still gained 10.3% last week, its biggest weekly win since 2009. The Dow Jones Industrial Average's 12.8% weekly gain was its biggest since 1938. But the market is still down 25% from the peak it reached a month ago.
The pandemic relief bill approved by the Congress and signed Friday by President Donald Trump includes direct payments to households, aid to hard-hit industries like airlines and support for small businesses. Analysts expect markets to remain turbulent, however, until the outbreak begins to wane.
"Sentiment once again took a turn for the worse going into a week of reckoning by means of economic fundamentals," Jingyi Pan of IG said in a commentary. "The rally seen for Wall Street last week may amount to little more but a relief rally with sentiment turning sour once again going into a fresh week."
The push to deliver financial relief has gained urgency worldwide as the outbreak widens. The number of cases in the U.S. has now surpassed those in China and Italy, climbing to more than 142,000 known cases, according to Johns Hopkins University. The worldwide total has topped 723,000, and the death toll has topped 34,000, while nearly 152,000 have recovered.
For most people, the new coronavirus causes mild or moderate symptoms, such as fever and cough that clear up in two to three weeks. For some, especially older adults and people with existing health problems, it can cause more severe illness, including pneumonia, or death.
The damage to corporate profits, the ultimate driver of stock prices, remains uncertain. Very few companies have dared to issue forecasts capturing the damage, though traders are girding for discouraging results in the next few weeks as earnings reporting season begins. Many companies have simply withdrawn their profit forecasts altogether.
At the start of this year, analysts expected S&P 500 companies' earnings would grow 4.4% in the January-March quarter. They now expect earnings will be down 4.1%, according to FactSet.
Earnings for airlines, which have been hit by lost bookings as businesses and individuals canceled travel plans to minimize their risk of contracting the virus, are expected to be catastrophic. Delta went from an expected 2.2% decline to a 108% plunge.
Energy companies have suffered, meanwhile, by a plunge in oil prices partly due to a price war that broke out early this month between Saudi Arabia and Russia. The energy sector of the S&P 500 has lost half its value this year.
U.S. benchmark crude dropped 4.2% or $91 cents on Monday to $20.60 per barrel in electronic trading on the New York Mercantile Exchange. It slid 4.8% to close at $21.51 a barrel on Friday. Goldman Sachs has forecast that it will fall well below $20 a barrel in the next two months because storage will be filled to the brim and wells will have to be shut in.
Brent crude, the international standard, gave up 4.2% or $1.16 to $26.79 per barrel.
The yield on the 10-year Treasury slipped to 0.67% from 0.68% late Friday. Lower yields reflect dimmer expectations for economic growth and greater demand for low-risk assets.
In currency trading, the dollar was at 107.75 Japanese yen, down from 107.94 late Friday. The euro weakened to $1.1078 from $1.1142.