By Anna Isaac and Chong Koh Ping

U.S. stock futures rose Monday following last week's sharp rally as investors assessed whether there may be a second wave of coronavirus infections with more lockdowns easing, and the implications for the economic recovery.

Futures tied to the Dow Jones Industrial Average gained 0.9%. The benchmark for blue-chip stocks jumped 3.2% on Friday after data showed the U.S. unexpectedly added 2.5 million jobs in May.

"The nonfarm payrolls have suggested that the depth and the nature of the recession may be less severe than first thought," said Edward Park, deputy chief investment officer at Brooks Macdonald.

"Even though the market sentiment has improved, there's a little bit of recognition that while the U.S. payroll figures were strong, assuming that we're well on the road to a full recovery is a little premature," he said.

The risk of a second wave of infections was likely to constrain investors' positivity. Some U.S. states are reporting an increase in the number of new infections after having lifted restrictions on social and business activity. California, Utah, Arizona, North Carolina, Florida, Arkansas and Texas, among others, have all logged rises in confirmed cases, according to a Johns Hopkins tabulation of a five-day moving average from over the weekend.

New York City, the U.S. area hit hardest by the pandemic, plans to begin reopening its economy Monday.

"The virus is not going to be gone in two months time," said Oliver Jones, senior markets economist at Capital Economics. "The sectors that are vulnerable to lockdowns such as transport still show investors expect a slow or painful withdrawal from the measures in coming months."

European stock indexes wavered between mild gains and losses. The pan-continental Stoxx Europe 600 ticked 0.2% lower. A fiscal stimulus plan proposed by the European Union's executive arm in recent weeks, which is likely to be discussed by top EU leaders on Thursday, faces resistance from some nations.

"There's still some lingering concern over the European Recovery Fund," said Dean Turner, economist at UBS Global Wealth Management. "As impressive and as needed as the recovery fund is, none of it comes alive until next year."

Investors may get fresh cues on the European Central Bank's response to the coronavirus pandemic and wider monetary policy when ECB President Christine Lagarde speaks to EU lawmakers at 9:45 a.m. ET.

Brent crude, the global gauge of crude-oil prices, edged down 0.2% to $42.22 a barrel after the Organization of the Petroleum Exporting Countries and its allies agreed to collective cuts of 9.6 million barrels a day until the end of next month. The gauge for U.S. crude fell 0.3%, after earlier crossing the $40-a-barrel threshold for the first time since early March.

Brent crude, the global gauge of crude-oil prices, edged down 0.2% to $42.22 a barrel after a press conference following an agreement by the Organization of the Petroleum Exporting Countries and its allies to collective cuts of 9.6 million barrels a day until the end of next month. The gauge for U.S. crude fell 0.3%, after earlier crossing the $40-a-barrel threshold for the first time since early March.

An emphasis on the need for strict compliance with the new production cuts may have prompted crude prices to give up their gains, said Bjarne Schieldrop, an analyst at SEB Markets.

Energy companies rallied ahead of the opening bell in New York. Marathon Oil Corporation rose 24% premarket, while Occidental Petroleum Corp. climbed 19%. Energy companies and airlines continued to lead gains ahead of market open, after a strong performance on Friday.

The yield on the benchmark 10-year U.S. Treasury note ticked up to 0.907%, from 0.903% Friday. U.S. yields surged at the end of last week to their highest level since late March after Friday's better-than-expected jobs report, signaling a significant shift in sentiment among bondholders who had been concerned about the prospects for the economy.

In Asia, Japan's Nikkei 225 added 1.4% by the close of trading, while the Shanghai Composite and South Korea's Kospi Composite posted more tepid gains. Markets in Australia were closed for a holiday.

"The strong set of labor data Friday gave more confirmation that an economic recovery is ongoing," said Eli Lee, head of investment strategy at Bank of Singapore.

Dollar weakness is likely to persist for the next six to 12 months, benefiting stocks in Europe and the emerging markets in Asia, he said. However, he remains cautious about betting on a sharp recovery, given uncertainties around the U.S. presidential election, escalating tensions between the U.S. and China, and a possible second wave of coronavirus infections.

Investors are weighing encouraging U.S. employment figures against worse-than-expected Chinese trade statistics, said David Chao, global market strategist for Asia Pacific ex-Japan at Invesco.

Data over the weekend showed that China's exports fell 3.3% in May from a year earlier, while imports fell 16.7%.

"This shows that consumption is still quite weak in China," said Mr. Chao, pointing to other suggestions of economic fragility, such as estimates that as many as 80 million workers were out of work earlier this year. Lower commodity prices also helped push imports down.

Write to Anna Isaac at anna.isaac@wsj.com and Chong Koh Ping at chong.kohping@wsj.com