Brent crude was up 6 cents or 0.1pc at $56.40 a barrel by 0911 GMT, and has risen 3.4pc since last Friday, its first weekly increase in six weeks.
U.S. West Texas Intermediate (WTI) was 3 cents or 0.1pc higher at $51.45 a barrel, up 2.2pc for the week.
"Oil prices appear to have stabilised this week on optimism that OPEC+ will once again do whatever it takes to tighten output and on hope that the coronavirus peak is nearing," said Edward Moya, senior market analyst at OANDA in New York.
In response to the demand slump, the Organization of the Petroleum Exporting Countries (OPEC) and its allied producers, a grouping known as OPEC+, are considering cutting output by up to 2.3 million barrels per day.
More than 1,350 people have died from the coronavirus in China, which has disrupted the world's second largest economy and shaken energy markets. Brent has fallen 15pc since the beginning of the year.
However, market sentiment improved as factories in China started to reopen and the government eased its monetary policy.
The World Health Organization also reassured traders by saying the big jump in China's reported cases reflected a decision by authorities to reclassify a backlog of suspected cases, and did not necessarily indicate a wider epidemic.
The International Energy Agency (IEA) said that first-quarter oil demand is set to fall versus a year earlier for the first time since the financial crisis in 2009 because of the coronavirus outbreak.
"Given the strength seen in the market this week, it suggests participants were factoring in even larger demand hits as a result of COVID-19 (the coronavirus)," said ING analyst Warren Patterson.
Some officials and analysts were still hopeful that the demand impact would remain limited to China.
"Our baseline thesis remains that oil demand destruction remains largely a China story and has yet to spill over to impact global demand," said Helima Croft, head of commodity strategy at Citadel Magnus.
U.S. Energy Secretary Dan Brouillette also told Reuters the coronavirus epidemic in China has had a marginal impact on energy markets and is unlikely to dramatically affect oil prices even if Chinese demand falls by 500,000 barrels per day.