Brent crude fell by 9 cents, or 0.1pc, to $70.02 a barrel by 1245 GMT, with prices repeatedly veering above and below $70 in choppy trading.
US West Texas Intermediate (WTI) was up 47 cents, or 0.8pc, at $59.10. US crude futures were trading for the first time since Friday after a long holiday weekend.
Investors, however, are concerned that the trade war between the United States and China could hit the global economy and dent fuel consumption.
Brent futures last week registered a decline of 4.5pc and WTI was slid by 6.4pc for its biggest weekly loss since December.
"Oil prices lack direction because the oil market currently finds itself caught between supply risks and concerns about demand," Commerzbank said in a note.
"A whole host of poor economic data from the major economic areas of the US, China and Europe, plus the entrenched situation in the trade talks, are not good news for the demand outlook."
On the flip-side, crude has gained support from supply cuts by the Organization of the Petroleum Exporting Countries (OPEC) and its allies since the start of the year, with political tensions in the Middle East another bullish influence.
No political solution appears forthcoming to end US sanctions that have largely taken Iranian and Venezuelan crude out of global markets.
"Brent is likely to resume its upward trend in line with its fundamentals, which are tight," said Harry Tchilinguirian, global oil strategist at BNP Paribas in London.
"This tightness is reflected in the generic front-month Brent futures time-spread. Backwardation is very steep at $1.33 a barrel - the last time we sustained such deep backwardation was in 2013, when spot Brent was trading above $100 a barrel."
Backwardation, a market structure where the spot price is higher than the price of crude in later months, tends to indicate tight supplies and a drawdown in inventories.
OPEC and allies including Russia are due to meet over June 25-26 to discuss output policy, but it remains unclear whether their production pact will be extended.
"Saudi Arabia and its Gulf allies will be under intense pressure to increase supplies," said Abhishek Kumar, head of analytics at Interfax Energy in London.
"Questions are also being raised over the unity of OPEC+ as Russia becomes increasingly concerned about losing its market share of oil exports to the US.