By Devika Krishna Kumar

NEW YORK (Reuters) - Oil steadied on Monday as hopes for a pick-up in demand later this year helped arrest last week's broad sell-off, but prices stayed under pressure as new European coronavirus lockdowns made a quick recovery look less likely.

Brent crude was up 6 cents or 0.1% to $64.59 a barrel by 1:33 p.m. ET (1733 GMT), while U.S. oil for delivery in April fell 22 cents, or 0.4%, to $61.20 a barrel ahead of expiry.

The more active U.S. crude futures for delivery in May rose 14 cents or 0.2% to $61.58 a barrel.

Both contracts fell more than 6% last week after making steady gains for months on the back of output cuts and an expected demand recovery.

"Oil (had) its worst week this year as concerns grow over a flaring up in COVID-19 cases across Europe," Dutch bank ING said in a note. "This comes at a time when there are clear signs of weakness in the physical oil market."

Physical markets have come under pressure as refiners around the world, including China and the United States, begin maintenance activities.

Chinese refinery maintenance season is due to peak in May and begin tapering in June, traders have said, depriving some crude grades such as those in West Africa of their main outlet. [CRU/WAF]

Nearly a third of French people entered a month-long lockdown on Saturday, while Germany plans to extend its lockdown into a fifth month, according to a draft proposal.

British Prime Minister Boris Johnson warned on Monday that the third wave of COVID-19 infections sweeping across Europe could be heading towards Britain.

"Vaccination campaigns haven't been as fast as the market had hoped for and consequently this will have an effect on the oil demand recovery, which in turn hurts prices," said Louise Dickson, oil markets analyst at Rystad Energy.

While a broad economic recovery remains elusive, Saudi Aramco Chief Executive Amin Nasser was optimistic on longer-term prospects for the world's top oil exporter.

On Sunday Nasser said global oil demand was on track to reach 99 million barrels per day (bpd) by the end of 2021.

"While I think demand is going to improve further as more economies ease travel restrictions in the coming months, the impact of this will be offset to some degree by rising oil supply," Fawad Razaqzada, market analyst at ThinkMarkets said.

"OPEC+ will be easing supply restrictions slowly, while U.S. shale production is likely to ramp up due to the attractive oil prices again. All told, I can’t see oil prices rising significantly further."