France's CAC 40 rose 0.1% in early trading to 4,426.13. while Germany's DAX fell 0.7% to 10,341.40. Britain's FTSE 100 edged less than 0.1% lower to 5,767.32.
US shares were set to drift lower with Dow industrials and S&P 500 futures almost unchanged.
The US House was expected to vote on Thursday on a USD 483-billion proposal to deliver more loans to small businesses and aid to hospitals after the Senate approved it on Tuesday.
The new bill would come on top of more than USD 2 trillion in aid that Congress has already approved. That, plus massive support for markets from the Federal Reserve, has helped the S&P 500 rise 25% since a low in late March. The index has roughly halved its loss from its record set in February.
Investors are still bracing for a severe, painfully deep recession after businesses shut down worldwide in hopes of slowing the spread of the coronavirus. Even as depressing economic and health reports pile up, some investors are looking ahead to parts of the economy reopening as infections level off in some areas.
Trading was mixed in Asia, with Japan's Nikkei 225 adding 1.5% to finish at 19,429.44, while South Korea's Kospi rose nearly 1.0% to 1,914.73.
Australia's S&P/ASX 200 gave up earlier gains, losing nearly 0.1% to 5,217.10. Hong Kong's Hang Seng added 0.4% to 23,977.32, while the Shanghai Composite inched down 0.2% to 2,838.50.
Prakash Sakpal and Nicholas Mapa, economists at ING, said a rise in oil prices, which have been crashing, was a help.
"But rising Covid-19 cases in the region will likely cap any rally," they said in a commentary.
Benchmark crude added USD 1.03 to USD 14.81 a barrel in electronic trading on the New York Mercantile Exchange. The price of a barrel of US oil to be delivered in June jumped 19% to settle at USD 13.78 on Wednesday.
It had zigzagged in the morning before turning higher after President Donald Trump threatened the destruction of any Iranian gunboats that harass US Navy ships, raising the possibility of a disruption to oil supplies.
The big gain, though, means it's recovered just a fraction of its steep losses. It was close to USD 30 at the start of last week and nearly USD 60 at the beginning of the year.
A collapse in demand for energy combined with continued production in countries around the world means too much oil is sloshing around, depressing its price.
Brent crude, the international standard, rose USD 1.16 to USD 21.53 per barrel.
The dollar fell to 107.62 Japanese yen from 107.71 yen on Wednesday. The euro fell to USD 1.0796 from USD 1.0822.